Mathematics From the Birth of Numbers (z-library.sk, 1lib.sk, z-lib.sk)

$Foreword: \$
4.7 Reliability of Digits and Calculations
9.5 Root, Exponential, and Logarithmic Equations
11.6 Euclidean and Non-Euclidean Geometries
13.1 Fundamental Trigonometric Functions
17.2 Anti-Snowflake and Anti-Square Curves
17.4 Sierpinski Triangle, Carpet, and Sponge
18.5 Cofactors and the Inverse of a Matrix
18.6 Solving Systems of Linear Equations
20 INTRODUCTION TO DIFFERENTIAL CALCULUS
20.2 Differentiating Algebraic Functions
20.3 Differentiating Transcendental Functions
20.4 Special Techniques of Differentiation
22.5 The Riemann Zeta Function and Hypothesis
26.4 Arc Length in a Polar Coordinate System
27.3 Center of Mass of Solids of Revolution
29.3 Solids Generated by Area Bounded by Two Curves
29.6 Solids with Known Cross-Section Areas
29.7 Transforming Double Integrals from Orthogonal to Polar Coordinates
30.2 Differentiating Distance and Velocity
30.3 Integrating Acceleration and Velocity
30.4 Velocity Vectors and Acceleration Vectors
34.2 First-Order Ordinary Differential Equations
34.4 Second-Order Ordinary Differential Equations
ISBN: 0-393-04002-X
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Mathematics
from the (Birth of 9{um6ers
2005-11

Mathematics
from the Birth of 9\[umSers
Jan Gullberg
Technical Illustrations
Par Gullberg
W • W • NORTON & COMPANY
New York
London

Copyright © 1997 by Jan Gullberg
All rights reserved
Printed in the United States of America
Art and cartoons: AnnaGreta Nordvall
Decorations and cartoons: Ann Gullberg, Kamen, and Kalin
Camera-ready copy for this book was produced entirely by the author, utilizing a
combination of modern desktop-publishing and traditional paste-up methods. The
text was written on an Apple Macintosh® Plus computer with Microsoft® Word
3.0 and 4.0; fonts used included New Century Schoolbook, Symbol, ITC Zapf
Cancery®, and ITC Zapf Dingbats®. Technical illustrations were generated with
a Macintosh® computer. Software used for the illustrations included Adobe
Illustrator®, Aldus FreeHand®, CA-Cricket® Draw™, Claris® CAD, and
Mathematica®. The three-dimensional, opaque fishnet graphics were rendered
with software designed by Par Gullberg. The main text of the manuscript was
produced on a Hewlett-Packard LaserJet 4 MP 600 DPI printer; for technical
illustrations an Apple LaserWriter Plus 300 DPI printer and a LaserJet 4 MP 600
DPI printer were used.
The nontechnical illustrations are a mixed bag of work by professionals and
nonprofessionals: original works include two cartoons by Leif Ekerling and
various illustrations by AnnaGreta Nordvall; reproductions of copyrighted cartoons
were obtained from sources specified in the text; the greatest number of
illustrations were made by the home-based artist Ann Gullberg; some illustrations are
works by Kamen and Kalin, who during their participation in the project aged
from nine to ten.
Library of Congress Cataloging-in-Publication Data
Gullberg, Jan.
Mathematics: froni the birth of numbers I Jan Gullberg
p. cm.
Includes bibliographical references and index.
ISBN 0-393-04002-X
1. Mathematics—History. I. Title.
QA21.G78 1996
510' .9—dc20
96-13428
CIP
W W Norton & Company, Inc., 500 Fifth Avenue, New York, N.Y. 10110
http :/7web. wwnor ton. com
W W Norton & Company Ltd., 10 Coptic Street, London WC1A 1PU
67890

To Per-Ota !Asflund, a friend in deed

VI
Contents in Brief
Contents in Brief vi
Preface vii
Contents i x
Foreword: "Mathematics in Our Culture" by Peter Hilton xvii
Information for the Reader xxiii
Chapter
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
Numbers and Language
Systems of Numeration
Types of Numbers
Cornerstones of Mathematics
Combinatorics
Symbolic Logic
Set Theory
Introduction to Sequences and Series
Theory of Equations
Introduction to Functions
Overture to the Geometries
Elementary Geometry
Trigonometry
Hyperbolic Functions
Analytic Geometry
Vector Analysis
Fractals
Matrices and Determinants
Embarking on Calculus
Introduction to Differential Calculus
Introduction to Integral Calculus
Power Series
Indeterminate Limits
Complex Numbers Revisited
Extrema and Critical Points
Arc Length
Centroids
Area
Volume
Motion
Harmonic Analysis
Methods of Approximation
Probability Theory
Differential Equations
Bibliography
Name Index
Subject Index
Symbols in
Common Use
1
31
69
97
183
215
231
263
295
335
363
385
457
533
547
599
625
637
671
683
721
765
779
787
797
827
837
851
871
891
907
923
961
989
1041
1053
1060
1092

Vll
Preface
My goal during the ten years that I have been writing this text has been
to bring together the history of mathematics with a broad treatment of its
foundations.
Three concurrent events led me to this pursuit:
o I had started preparing a translation and revision of a text I had
written on units and measurements that I thought could benefit from
a 10- to 12-page introductory mathematics text.
o My son studied mathematics as part of an engineering degree, and
mathematics became the common subject of our conversations,
making the intended introductory text snowball.
0 As a physician, I had become disillusioned with an ever-increasing
number of "studies" in which medical and other life science matters
were proved and disproved and then again, in a cyclic manner,
proved, disproved, proved ... with "irrefutable mathematical
support"; this use of mathematics strengthened my desire to enjoy
mathematics where it was not brutalized.
In the beginning, I thought I would be able to gain something useful for
my units and measurements text (now relegated to a locked drawer),
but I mainly wrote for myself- as a means of exploring mathematics
and its history more completely than I ever had before.
After focusing on numbers and their symbols, my quest continued
from the four fundamental rules of arithmetic to calculus and its
culmination in differential equations. Along the way, I delved into
algebra, the theory of functions, geometry, trigonometry, hyperbolic
functions, and analytic geometry, and also followed trails to number
theory, symbolic logic, set theory, Boolean algebra, transfinite
numbers, topology, fractals, vector analysis, and probability theory.
My draft text was later polished with the help of distinguished
professional mathematicians, linguists, and historians. In its final form,
1 hope this format of mathematics and its history will appeal to readers
who have, or would like to develop, a fondness for mathematics.
Among those who have given me constructive criticism, advice,
suggestions, and contributions to the text, Johan C. Martensen-
bookseller, specialist in technical and scientific publications - has
been available throughout the project; his never-ending enthusiasm,
encouragement, and wealth of ideas were an enormous asset. I thank
Johan and all other individuals wholeheartedly for their help, and ask
pardon if I have not always been fully receptive to their wise counsel
and erudition.
My thanks are extended to librarians in Sweden, Norway, and the
United States for their kind help and efforts in procuring originals and
facsimiles of old texts.
I am delighted that Peter Hilton adorns my work with a foreword:
"Mathematics in Our Culture".
The knowledge and persistence of my editor, Joseph Wisnovsky,
have aided the project enormously. Not only did he marshal the
resources of W. W. Norton, but also went through the entire text himself.
I began writing this text in Angola, Indiana, in November 1986
and continued in Forks, Sequim, and Moses Lake, Washington; in
Karlskoga, Sweden; in Rjukan, Norway; in Moab, Utah. The final
design was completed in June 1996 in Mosj0en, Norway, and I checked
the galley proofs in Morton, Washington, in the autumn of 1996. My
peripatetic lifestyle has been crucial to the gathering of data and
development of ideas.

Vlll
Exploring Mathematics ...
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Exploring mathematics is rather like walking through a Troy-town
labyrinth, with its path oscillating between the periphery and the
center. Spreading out into numerous branches, mathematics is still
a wholeness and its core has central themes and goals.
The name Troy-town, or Walls of Troy alludes to the defense
walls around the ancient city of Troy (or cities as, in fact, nine cities
of Troy have been excavated, one on top of the other). The origin of the
Troy-town design is shrouded in mystery and might even antedate
the fabled cities.
The oldest Troy-town designs known are carved into the stone
wall of a 5000-year-old grave in Sicily. Similar designs are
scattered around the world. Pregnant women in Java, Sumatra, and
India have long used the pattern to search for peace in meditation; in
the United States, Navajo Indians see the pattern as a model of the
world's creation. The greatest number of Troy-town labyrinths
outlined in stone have been found in Sweden, over 300, mainly along its
eastern coastline and on Gotland, in the Baltic. The above design is
from a Troy-town outside the medieval town of Visby on Gotland; the
actual labyrinth is 19 meters wide.
One cannot get lost in a Troy-town labyrinth. At the endpoint one
rests or, according to Nordic folklore, finds one's heart's desire.

IX
Contents
Page
Contents in Brief vi
Preface vii
Contents i x
Foreword: "Mathematics in Our Culture** by Peter Hilton xvii
Information for the Reader xxiii
Chapter
1 NUMBERS AND LANGUAGE 1
1.0 The Origins of Reckoning 3
1.1 Numbers and Numerals 5
1.2 Number Names 7
1.3 Etymology of English Number Names 26
1.4 Numbers vs. Infinity 29
2 SYSTEMS OF NUMERATION 31
2.0 Forms of Notation 32
2.1 Additive Notation 34
2.2 Multiplicative Notation 44
2.3 Positional Notation 46
2.4 Decimal Position System 48
2.5 Sexagesimal Numeration 56
2.6 Vigesimal Numeration 58
2.7 Duodecimal Numeration 61
2.8 Binary, Octal, Hexadecimal 62
2.9 Special Forms of Notation 66
3 TYPES OF NUMBERS 69
3.0 An Expanding Universe of Numbers 70
3.1 Rational Numbers 72
3.2 Prime Numbers 77
3.3 Perfect and Amicable Numbers 82
3.4 Irrational Numbers 84
3.5 Imaginary and Complex Numbers 87
3.6 The Quest for n 89

X
CONTENTS
4 CORNERSTONES OF MATHEMATICS 97
4.0 Beginnings 98
4.1 Symbols Galore 101
4.2 Fundamental Operations 113
4.3 Laws of Arithmetic and Algebra 132
4.4 Powers and Roots 134
4.5 Logarithms 150
4.6 Mathematical Proof 157
4.7 Reliability of Digits and Calculations 161
4.8 Simple Calculating Devices 168
5 COMBINATORICS 183
5.0 Historical Notes 184
5.1 Multiplication Principle 186
5.2 Permutations 188
5.3 Combinations 196
5.4 Samples with Replacement 199
5.5 Graph Theory 201
5.6 Magic Squares and Their Kin 205
6 SYMBOLIC LOGIC 215
6.0 Historical Notes 216
6.1 Pitfalls 219
6.2 Propositions 220
6.3 Tautologies 225
6.4 Syllogisms and Proofs 227
6.5 Logic Circuits 229
7 SET THEORY 231
7.0 Introduction 232
7.1 Sets and Their Contents 233
7.2 Venn Diagrams 242
7.3 Algebra of Sets 249
7.4 Boolean Algebra 252
7.5 Transfinite Numbers 257
8 INTRODUCTION TO SEQUENCES AND SERIES 263
8.1 Terminology 264
8.2 Finite Sequences and Series 266

CONTENTS
xi
8.3 Infinite Series 270
8.4 The Tower of Hanoi 285
8.5 The Fibonacci and Related Sequences 286
8.6 Figurate Numbers 289
9 THEORY OF EQUATIONS 295
9.01 History 297
9.02 Groundwork 302
9.1 Linear Equations 307
9.2 Equations with Absolute Values 308
9.3 Quadratic Equations 309
9.4 Inequalities 312
9.5 Root, Exponential, and Logarithmic Equations 313
9.6 Cubic Equations 316
9.7 Quartic Equations 320
9.8 Systems of Equations 325
9.9 Diophantine Equations 330
10 INTRODUCTION TO FUNCTIONS 335
10.0 Historical Notes 336
10.1 Groundwork 336
10.2 Elementary Functions 348
10.3 Continuity and Limits 355
11 OVERTURE TO THE GEOMETRIES 363
11.0 History 364
11.1 Geometric Abstraction 370
11.2 Perspective and Projection 372
11.3 Form and Shape 376
11.4 Survey of Geometries 377
11.5 Topology 378
11.6 Euclidean and Non-Euclidean Geometries 381
12 ELEMENTARY GEOMETRY 385
12.0 What Do We Mean by "Elementary Geometry"? 386
12.1 Geometric Elements and Figures 386
12.2 Units of Measurement 409
12.3 Euclidean Construction 413
12.4 Theorems and Formulas 425
12.41 Plane Geometry 426
12.42 Solid Geometry 446

Xll CONTENTS
13 TRIGONOMETRY 457
13.0 Scope and History 458
13.1 Fundamental Trigonometric Functions 470
13.2 Inverse Trigonometric Functions 477
13.3 Solving Triangles 478
13.4 Graphs, Domains, Ranges 502
13.5 Trigonometric Identities 507
13.6 Trigonometric Equations 516
13.7 Limits 528
14 HYPERBOLIC FUNCTIONS 533
14.0 Introduction 534
14.1 Fundamental Hyperbolic Functions 536
14.2 Inverse Hyperbolic Functions 539
14.3 Identities 541
15 ANALYTIC GEOMETRY 547
15.0 Scope and History 548
15.1 Rectilinear Figures 549
15.2 Conic Sections 559
15.3 Shifting Orthogonal Coordinates 572
15.4 Polar Coordinate Systems 576
15.5 Parametric Equations 586
16 VECTOR ANALYSIS 599
16.0 Scope and History 600
16.1 Basic Vector Algebra 602
16.2 Scalar and Vector Components 607
16.3 Multiplication of Vectors 614
17 FRACTALS 625
17.0 What Are Fractals? 626
17.1 The Snowflake Curve 627
17.2 Anti-Snowflake and Anti-Square Curves 629
17.3 The Cantor Set 630
17.4 Sierpinski Triangle, Carpet, and Sponge 631
17.5 The Mandelbrot Set 633
17.6 The Dimension Concept 635

CONTENTS
• • •
Xlll
18 MATRICES AND DETERMINANTS 637
18.0 Scope and History 638
18.1 Matrices - Presentation 640
18.2 Matrices - Rules of Operation 642
18.3 Determinants 646
18.4 Special Matrices 654
18.5 Cofactors and the Inverse of a Matrix 659
18.6 Solving Systems of Linear Equations 661
19 EMBARKING ON CALCULUS 671
19.1 What Is Calculus? 673
19.2 History 674
20 INTRODUCTION TO DIFFERENTIAL CALCULUS 683
20.1 Derivatives and Differentials 685
20.2 Differentiating Algebraic Functions 689
20.3 Differentiating Transcendental Functions 695
20.4 Special Techniques of Differentiation 705
20.5 Partial Differentiation 710
20.6 Mean-Value Theorems 716
21 INTRODUCTION TO INTEGRAL CALCULUS 721
21.1 Basic Concepts 722
21.2 Methods of Integration 723
21.3 The Definite Integral 747
21.4 Multiple Integrals 753
21.5 Improper or Unrestricted Integrals 756
22 POWER SERIES 765
22.1 Convergence 766
22.2 Taylor's and Maclaurin's Series 767
22.3 Expanding Transcendental Functions 771
22.4 Binomial Expansion 775
22.5 The Riemann Zeta Function and Hypothesis 778
23 INDETERMINATE LIMITS 779
23.1 A Retrospect 781
23.2 L'Hospital's Rule 782

xiv
CONTENTS
24 COMPLEX NUMBERS REVISITED 787
24.1 Introduction 788
24.2 Sums and Differences 789
24.3 Products and Quotients 790
24.4 Powers 791
24.5 Roots 793
24.6 Logarithms 794
25 EXTREMA AND CRITICAL POINTS 797
25.1 One Independent Variable 798
25.2 More than One Independent Variable 815
25.3 Functions with Restrictions 822
26 ARC LENGTH 827
26.1 Basic Principle 829
26.2 The Catenary 831
26.3 Arc Length in Parametric Form 832
26.4 Arc Length in a Polar Coordinate System 835
27 CENTROIDS 837
27.1 Mass Point Systems 838
27.2 Plane Figures and Laminas 842
27.3 Center of Mass of Solids of Revolution 847
28 AREA 851
28.1 Plane Surfaces 852
28.2 Surf ace of Revolution 860
28.3 Work 867
29 VOLUME 871
29.0 Introduction 873
29.1 Disk Method 873
29.2 Shell Method 877
29.3 Solids Generated by Area Bounded by Two Curves 880
29.4 Translation of Axes 882
29.5 Guldin's Second Rule 884
29.6 Solids with Known Cross-Section Areas 885
29.7 Transforming Double Integrals from Orthogonal to Polar
Coordinates 887

CONTENTS
XV
30 MOTION 891
30.1 Laws of Kepler and Newton 893
30.2 Differentiating Distance and Velocity 896
30.3 Integrating Acceleration and Velocity 900
30.4 Velocity Vectors and Acceleration Vectors 901
30.5 Space-Time; Mass and Energy 904
31 HARMONIC ANALYSIS 907
31.0 Historical Notes 909
31.1 Fourier Series 910
31.2 Expanding Discontinuous Functions 915
31.3 Expanding Even or Odd Functions 918
32 METHODS OF APPROXIMATION 923
32.1 Negligible Terms 925
32.2 Interpolation 926
32.3 Graphic and Iterative Methods 931
32.4 Numerical Integration 947
33 PROBABILITY THEORY 961
33.0 Introduction 962
33.1 History 963
33.2 The Basics 966
33.3 The Probability Density Function 971
33.4 Central Tendency 974
33.5 Dispersion 976
33.6 Normal Distribution 979
34 DIFFERENTIAL EQUATIONS 989
34.1 Fundamental Concepts 991
34.2 First-Order Ordinary Differential Equations 994
34.3 Formulating Differential Equations 1000
34.4 Second-Order Ordinary Differential Equations 1015
BIBLIOGRAPHY 1041
Works Cited 1048
NAME INDEX 1053
SUBJECT INDEX 1060
SYMBOLS IN COMMON USE 1092

xvii
Foreword: Mathematics in Our Culture
The unstated premise of this book - a premise that virtually all
mathematicians would agree to - is that mathematics, like
music, is worth doing for its own sake. The author is, by
profession, a medical man, but he has a love of mathematics
and wants others to share his enthusiasm.
This is not to deny the great usefulness of mathematics;
this very usefulness, however, tends to conceal and disguise
the cultural aspect of mathematics. The role of music suffers
no such distortion, for it is clearly an art whose exercise
enriches composer, performer, and audience; music does not
need to be justified by its contribution to some other aspect of
human existence. Nobody asks, after listening to a Beethoven
symphony, 'What is the use of that?'. Moreover, mathematics
does not gain in utility by having its inherent worth ignored -
on the contrary, an appreciation of mathematics and an
understanding of its inherent quality and dynamic are
necessary in order to be able to apply it effectively.
The first serious error we often meet in considering the
role of mathematics is the confusion of education with
training. This error, of course, goes far beyond mathematics1
- our bureaucrats and politicians now use the two terms quite
synonymously - but it is particularly meretricious when
applied to mathematics. For students, and their parents,
believe that mathematics education should consist exclusively
of the acquisition of a set of skills that will prove useful in their
later careers; so the skills must be learned, that is, committed
to memory, and no real understanding need occur. Of course,
we cannot, in fact, predict what skills the student will need.
What we can predict is that those skills will change and that
the student will need to understand and not merely to
remember. Adaptability to change is itself a hallmark of
successful education, and it is change, not any specific
technology, that most aptly characterizes life today and in the
foreseeable future. A genuine education enables one to
acquire, for oneself, the skills one happens, at a given stage of
one's life, to need. A training, on its own, contributes almost
nothing to education and produces distressingly ephemeral
advantages.
The usefulness of mathematics leads to other, related
abuses. Since mathematics is useful, its acquisition must be
tested. Since, in the perverse view we are deprecating, it is a
skill, it is tested as a skill. Since it is useful, it must be taught
to all. Thus the testing problem becomes enormous, and
grading by machine becomes commonplace. The result is that
standard tests are applied that have almost nothing to do with
the acquisition of mathematical understanding and put a
premium on brute knowledge and memory, speed and slickness.
They provide no opportunity for the student to explain his or her
answer and treat all 'wrong' answers as equally wrong. They
are, in short, inimical to mathematics itself.
1 It is a wry commentary on the value-system in the United States that
one speaks there of 'teacher training' and 'driver education'!

xviii FOREWORD
Further, the study of mathematics starts with the teaching
of arithmetic, a horrible, wretched subject, far removed from
real mathematics, but perceived to be useful. As a result vast
numbers of intelligent people become 'mathematics avoiders'
even though they have never met mathematics. Their desire to
avoid the tedium of elementary arithmetic, with its boring,
unappetizing algorithms and pointless drill-calculations, is
perfectly natural and healthy.
To those intelligent people, it must seem absurd to liken
mathematics to music as an art to be savored and enjoyed even
in one's leisure time. Yet that is how it should appear and
could appear if it were playing its proper role in our (otherwise)
civilized society. Just as an appreciation of music is a
hallmark of the educated person, so should be an appreciation of
mathematics. The author of this book is a splendid example of
an educated person bearing this hallmark.
An Educated Person
There is, we claim, a valid and valuable concept of an
educated person. The ancient Greeks had this concept, and it
included for them an appreciation of mathematics, especially
geometry; on the other hand, the Romans, conspicuously, did
not. As Philip Howard wrote, reporting on the 1989 meeting of
the British Classical Association: The Romans were bad at
science. They were practical men who followed intellectual
pursuits only if they were useful and profitable, or, in the
uncharming vogue phrase, "bankable skills". It is an attitude
that is still with us.' Or perhaps one should say that it is again
with us. For the broader concept of education was current in
the 17th and 18th centuries in Britain and animated those who
founded the Royal Society of London; other nations, too, in
Europe and elsewhere in the world, had their Enlightenments,
their Renaissances. However, the concept began to undergo a
curious transition in Victorian England. Certainly, it
continued to connote the desire and the ability to go on
learning, by reading and other forms of study; and it implied
a familiarity with, and appreciation for, poetry, literature,
music, the arts, and architecture. However, when the
transition was complete, it carried two rather unfortunate
connotations as well. The term tended to be applied to members of the
leisured class (and, naturally but sadly, predominantly to the
masculine sex); and there was no implication of a knowledge
or appreciation of science.
This last feature largely persists to this day in the English
speaking countries. Exasperation with its manifestations led
C. P. Snow to deliver his celebrated Rede Lecture, The Two
Cultures', in which he deplored the prevalence, in positions of
prominence and influence, of people having no knowledge or
understanding of the Secorid Law of Thermodynamics. Of
course, Snow was not the first to remark on this phenomenon,
but his own popularity as a writer and reputation as a thinker
and man of affairs undoubtedly broadened the discussion, if

Peter Hilton: Mathematics in Our Culture
xix
it did not always succeed in deepening it. It is important to
recall that Snow's viewpoint that a person was only to be
considered educated if he or she was versed in the arts and the
sciences was by no means universally accepted at the time his
lecture was delivered and published.
The problem is very different in certain contemporary
societies where cultural philistinism is rampant. Now it is
often necessary to argue that a technologically advanced
society needs people with an understanding of history, an
appreciation of language (theirs and other people's) and an
awareness of the 'higher purposes' to which their increased
affluence and computerized efficiency give them access. The
tendency, to which we have already drawn attention, to
confuse education with training has led, at least in the English-
speaking world, to a marked down-grading of the study of the
arts and the humanities, and to the emergence of the
dangerous illusion that a modern industrial society should
encourage applied science at the expense of pure science. Such
an attitude, had it been widespread 30 years ago, would, for
example, have seriously impeded the development of 'medium
temperature' superconductors and laser technology. It is
surely clear, moreover, that an educated person should have
some understanding of both pure and applied aspects of
science. If, for example, he or she is to appreciate the actual
and potential roles of the computer in our and future societies,
then the educated person must appreciate significant parts of
science, technology, logic, and mathematics.
What Is Mathematics?
It is my special case that mathematics is common to the 'two
cultures', and that therefore the educated person should
appreciate it. However, it is not reasonable to expect lay
persons to understand the details of sophisticated
mathematical reasoning. Nevertheless, enough has surely been
said to imply that our educated person must appreciate the
role of mathematics in science and technology. Richard
Feynman, echoing the thought of Galileo, has said: 'Nature
talks to us in the language of mathematics', and it behoves
educated people to understand just what this profound
aphorism implies. Certainly such an understanding cannot
be achieved without a far better insight into what mathematics
actually is than is commonly found even among university-
trained people today. Yet such insight would not be enough;
for mathematics grows and develops in many ways unrelated
to science, and thus plays a crucial role in the history of
human thought. So I argue that the educated person must
understand what mathematics is - but not in the sense
of a dictionary definition. Such a person must have an
appreciation of mathematical reasoning and of the role of
mathematics in the evolution and development of human
society. Such an appreciation requires one to understand
something of what mathematicians do - this would provide a

FOREWORD
much better working description of what mathematics is, in
practice, than any dictionary could provide2.
Unfortunately very few people have this kind of
appreciation of the true nature of mathematics. The most common
fallacy, even among otherwise well-informed people, is, as we
have said, to confuse mathematics with elementary
arithmetic, and thus to suppose that progress in mathematics
consists of performing ever more complicated calculations
with speed, dexterity, and accuracy. Thus, for example,
Dustin Hoffman received an Oscar in 1989 for his portrayal of
the autistic brother in the film The Rain Man*. This person
is an 'idiot savant', capable of performing rapid, totally
unmotivated mental calculations such as computing 341 x 127,
or taking Vl9 to 10 decimal places. However, he is described
by various critics, in their reviews of the film, as a genius. It
is surely unnecessary for me to belabor the point further that
such an extraordinary ability, far from being evidence of
genius, is usually an indication of stupidity - as in this case.
There have been rare exceptions, such as the great German
mathematician, astronomer, and physicist Karl Friedrich
Gauss, the British civil engineer George Parker Bidder, and
the statistician A. C. Aitken. However, it is interesting and
significant to note that Gauss's powers of mental calculation
declined as his genius grew, thus testifying to the antithesis
between calculation and mathematical insight which we are
claiming to exist.
In real life a characteristic example of the idiot savant was
the Derbyshire agricultural laborer Jedediah Buxton, who was
able to demonstrate that the Fermat number 22 - 1 is not prime
by actually factorizing it when it was given to him in decimal
notation. He performed this feat in his head while carrying
out his everyday duties. Buxton was brought to London to
be examined by a group of Fellows of the Royal Society. He
was taken to the theater to see Garrick perform, to see how he
would react to the experience. He reacted by compulsively
counting the number of steps Garrick took during the
performance! Thus indeed did Buxton symbolically
demonstrate that arithmetical skill, of however high an order, is no
part of our culture.
This justified conviction, on the part of many sensitive
and 'educated' people, that arithmetic cannot be regarded as a
part of the individual's cultural equipment, together with the
erroneous belief that arithmetic is the essence of mathematics,
has led to the widely-held view that mathematics itself is not to
be regarded as a component of a liberal education. Thus many
aesthetes are to be found positively glorying in their ignorance
of, and ineptitude in, mathematics. Such people may proudly
announce that they do not understand railway timetables, and
are merely vexed by their difficulty in computing the tip in a
1 Bertrand Russell's famous dictum that 'mathematics is the subject
in which you don't know what you're talking about, and don't care
whether what you say is true' is merely a philosophical joke, though
a good one!

Peter Hilton: Mathematics in Our Culture
xxi
restaurant. There are not to be found educated people who
glory in their inability to use their language3 or to read
properly; anybody with such a difficulty would doubtless seek
to conceal it.
Genuine mathematics, then, its methods and its concepts,
by contrast with soulless calculation, constitutes one of the
finest expressions of the human spirit. The great areas of
mathematics - algebra, real analysis, complex analysis,
number theory, combinatorics, probability theory, statistics,
topology, geometry, and so on - have undoubtedly arisen from
our experience of the world around us, in order to systematize
that experience, to give it order and coherence, and thereby
to enable us to predict and perhaps control future events.
However, within each of these areas, and between these areas,
progress is very often made with no reference to the real world,
but in response to what might be called the mathematician's
apprehension of the natural dynamic of mathematics itself.
Mathematics, while essential to science, as the Nobel
prize-winning physicist Feynman has so vividly testified, has
its own internal dynamic, powerful and subtle. Often, and
today most especially, mathematics moves forward not under
the stimulus of science but under the stimulus of its own recent
advances. Applied mathematicians will often find a piece of
mathematics, developed for its own sake, the precise tool they
need for the expression and elucidation of their scientific
problem. And applied mathematicians, once they have
modeled the problem mathematically, proceed very much as
the pure mathematician would.
Thus it emerges that there is no great difference between
the procedures of pure and applied mathematics - there is
really only one mathematics. Of course there is the difference
that the source of the problem comes in one case from
mathematics itself and in the other from the real world; but
even here this difference is confined to the original source of
the problem - the applied mathematician grappling with a
differential equation is, at that point, behaving in a manner
indistinguishable from that of a pure mathematician. Indeed,
to strike a controversial note, it could be argued that the pure
mathematician has opportunities for application that
transcend those of the applied mathematician. One can apply
mathematics to solve problems in physics - but it is difficult
(though not, perhaps, absolutely impossible) to conceive of
applying physics to solve problems in mathematics. However,
within mathematics, it is perfectly clear, indeed
commonplace, that one may, for example, apply algebra to solve a
problem in geometry, or apply geometry to solve a problem in
algebra.
The foregoing discussion is designed to show, in outline,
what mathematics is. My own position, as a mathematician,
is to be suitably humble about my own contributions to
mathematics, but not modest at all about my claims for
6 Regrettably, statistical evidence is accumulating to indicate that
students, offered training rather than education, and fascinated by the
potential of modern technology, are increasingly unable to use their
language properly to convey their ideas.

FOREWORD
mathematics itself. This was the position adopted by my
teacher and friend, the great British topologist Henry
Whitehead - though he had far less justification for his
humility! Whitehead argued that there are relatively few
pursuits in life that are inherently worth while - he instanced
the making of music and the design of elegant and useful
furniture - and that doing, or at least appreciating,
mathematics is one of them. It is surely reasonable to equate
Whitehead's concept of intrinsically valuable pursuits with
our own concept of the desiderata of the educated person. There
is, in fact, no doubt in my mind that mathematical
appreciation is not only a component part of the education of civilized
people, but a pillar of that education. I long for the day when,
indeed, mathematics will be appreciated and enjoyed by
educated laymen as an art and also respected as the mainstay
of science. It has been so in the past, but it is not so now. Is it too
optimistic to hope it might be so again?
The present text affords solid grounds for believing it may
not be too optimistic. The author, imbued with the spirit that I
have tried to convey, has sought to bring to his readers an
understanding of the art and the science of mathematics.
That he is not a professional mathematician makes his
dedication all the more commendable - and remarkable. His
text has been scrutinized by a number of leading
mathematicians and judged to be of great merit in conveying both
the content and the spirit of a true mathematical education.
I hope it may enjoy the success it richly deserves and exert
the influence it should on present and future generations of
intelligent readers.
Peter Hilton
Distinguished Professor of Mathematics, Emeritus
State University of New York at Binghamton
Santa Clara, California, May 1996
Adapted from
"The Mathematical Component of a Good Education",
Miscellanea Mathematica, Springer-Verlag, Berlin, 1991.

xxiii
Information for the Reader
Although all of the problems in this book have solutions to
make it suitable for armchair reading, you may find it
worthwhile to have pencil and paper handy. You choose
whether to solve a given problem or prove a theorem, simply
read and accept, or, perhaps, return to it at another time. There
is often more than one way to find an answer or a proof- your
approach may be as good as the one in the text.
To avoid tedious work, you should have a good scientific
calculator.
• This mark denotes the beginning (left-hand side) of
problems and examples, and sometimes, for clarity, the end
(right-hand side).
The notations and symbols recommended by the
International Standards Organization (ISO) are used throughout,
with one exception: a decimal point is used instead of a
decimal comma.
Instead of the American and British practice of separating
digits with commas,
10,521 1,345,876 10,521.3672389124
in this text, following the recommendations of ISO, digits are
marked off in groups of three by spaces, working in both
directions from the decimal point:
10 521 1345 876 10 521.367 238 912 4
Greek proper names are in their customary English
versions within the text, and in transliterated Greek in the
margin; e.g., Euclid within the text, Evcleidis in the margin.
Names originating in Cyrillic, Arabic, and Hebrew
alphabets may be difficult to locate in the index, as the English
transliterations of such names vary between sources and the
reader might be familiar only with a version not used in this
text; such inconsistencies and difficulties are even more
pronounced for Chinese and Japanese names.
You will not find descriptions of mathematicians'
personalities or detailed biographies in this text. Generally only a
mathematician's name, dates, nationality, and other fields of
scientific activity are given along with a short account of his
or her contribution to the topic under discussion. Dates are
given to show the time period in history or chronology of
events, and are often repeated for convenience and clarity.
Titles of knighthood or nobility are generally omitted.
Readers of earlier printings have kindly responded to my
invitation to comment on the text and so make it more
dependable. I would like to give special thanks to Dennis E.
Knopf, Herb Sachs, and Paul H. Stanford for their detailed
and savvy comments.
JG

1
Chapter
1
NUMBERS AND LANGUAGE
Page
1.0 The Origins of Reckoning 3
1.1 Numbers and Numerals 5
1.2 Number Names 7
1.3 Etymology of English Number Names 26
1.4 Numbers vs. Infinity 29

mm»
4'-: ■'
>:^iC'\ >:
*?'> s
Axel Ebbe (1868-1941):
"Breaking Away from the Darkness of Ignorance"
University of Lund, Sweden
... and then there was MATHEMATICS

3
The Origins of Reckoning
'Without words there is no possibility of meaning.
Thomas Hobbes, Leviathan (1651)
Scholars generally agree that our ability to count, and our
vocabulary of counting, arose to meet practical needs and
developed over many thousand years.
Numbers were originally expressed by reference to parts of the
human body - nose, eyes, ears, arms, hands, feet, and
particularly fingers and toes - in a specific order, making the
names of these organs and members double as names of
numbers.
These number names were eventually simplified in their
spoken forms - long before the advent of writing - as the
concept of abstract numeration came into being. Studies of
"primitive peoples", who live in isolation from civilization,
suggest several stages of development toward strict number
names.
Hunting and gathering peoples - such as the aborigines of
Australia, Tasmania, and Papua New Guinea - generally
have had, or still have, specific names only for the numbers
one and two and, sometimes, three, yet they can count to as
many as six by combining numbers, for instance like this:
1
2
3
4
5
6
one
two
two-one
two-two
two-two-
one
two-two-two
one
two
three
one-three
two-three
three-three
In these languages, speakers refer to everything beyond six as
many, much, or plenty - more plenty or less plenty, as the case
might be.
Analogies exist in contemporary English. When we count
trees, we refer to each tree, counting one, two, three ... seven,
eight... until the trees cease to be important, qua trees; we then
speak of a clump of trees, a coppice or copse, a grove, until we
find ourselves in a wood, and finally in a forest.
There are many other ways of expressing an undefined
number of, most commonly animate, objects: a gaggle of
geese, a school of fish, a swarm of bees, a pride of lions.
Similarly, many things are thought of, used, or counted in pairs, for
instance a brace of fowls, a brace of pistols, a brace of oxen.

Chapter 1 NUMBERS AND LANGUAGE
Dual, Trial, Quadrual
In many languages nouns and pronouns had, or still have,
more than two forms of number, in some cases up to five:
singular, dual (2), trial or trinal (3), quadrual (4), and plural.
Trial and quadrual forms are today found nearly
exclusively in Austronesian languages (spoken in Australia,
Polynesia, and Melanesia). A typical example of dual and
trial forms of personal pronouns is offered by the language of
one small island in the Melanesian archipelago:
Singular
Dual
Trial
Plural
1st person
2nd person
3rd person
ainjak (I) aijumrau (we two)
aiek (thou) aijaurau (ye two)
aien (he or she) arau (they two)
aijumtai (we three)
aijautaij (ye three)
ahtaij (they three)
aijam (we)
aijaua (ye)
ar (they)
In Sanskrit, the classical language of ancient India, nouns
had three forms: singular, dual, and plural. These are
current in many of its descendant languages and in several
unrelated languages:
Classical Greek
Arabic
Greenlandic (Iniipik)
Singular
(one man)
antropos
radjul
inuq
Dual
(two men)
antropo
radjulayn
innuq
Plural
(men, people)
antropoi
radjjaal
inuit
While Arabic has dual forms for practically all nouns,
Hebrew - also of the Semitic family of languages - uses the
dual form chiefly for paired body parts (two ears, two feet) or
objects consisting of two parts (tongs, scissors). Dual forms of
nouns are also found in Old and Middle Irish, and in older
forms of Russian.
Whereas Modern English and its precedent forms Old
English (c. 550 - c. 1100) and Middle English (c. 1100 - c. 1500)
lack dual forms in nouns, both Old English and Middle
English had dual forms in the 1st and 2nd persons of personal
pronouns, as in the table below for early Old English forms:
1st Person
Nominative
Genitive
Dative
Accusative
Singular
ic
mm
me
mec
(I)
(mine)
(to me)
(me)
Dual
wit
uncer
un
uncit
(we two)
(of us two)
(to us two)
(us two)
Plural
we
ure
us
usic
(we)
(our)
(to us)
(us)
2nd Person
Nominative
Genitive
Dative
Accusative
Singular
tiiu
thin
the
thee
(thou/you)
(thine/your)
(thee/to you)
(thee/you)
Dual
git
incer
inc
incit
(ye/you two)
(of you two)
(to you two)
(you two)
Plural
ge
eower
eow
eowic
(ye)
(your)
(to you)
(you)
th indicates a single sound as in Modern English then, either.
After the 13th century, all traces of the dual form in English
are gone.

1.1 Numbers and Numerals
vonLINNE Carl (1707-1778)
Swedish botanist; originator
of the scientific classification
of plants and animals; held
the first chair in Medicine at
Uppsala for a year (1741), then
held chair in Botany.
Although most often attributed
to Linne, the maxim Nomina
si nescis ... was originally
used by the Spanish scholar
Isidore of Seville (c. 560 - 636),
foremost encyclopedist,
composer of the encyclopedic
dictionary Etymologiae.
Cardinal Numbers
Ordinal Numbers
Latin, whole, entire;
literally untouched
Latin, frangere, "to break"
9\[pmina si nescis, perit et cognitio rerum.
Carl von Linne, Critica botanica (1737)
"Who knoweth not the names, knoweth not the subject."
The earliest traces of reckoning, uncovered by archaeological
excavations in the Middle East, date from about a hundred
centuries before our time. Some of these finds suggest a form
of counting where markers, counters, or tokens corresponded
directly with the things or goods they represented. From these
systems, symbols for abstract numbers and quantities
eventually developed. These abstract symbols applied to any
and every object, whereas symbols previously each
represented a specific commodity.
Thus, the mathematical concept of pure quantity was born.
Old English for "number" is rim, which also means "part";
Middle English got no(u)mbre via Old French from Latin
numerus, "number", traced to Greek vijuo, "distribute",
"share" - recognized in vijuecncj (nemesis), "allotment",
"portion" (as of Fate).
If mathematics is the language of science, the language of
mathematics is numbers, its grammar being represented by
the several kinds of numbers and by the ways in which we use
them.
The terms number and numeral may often be used
interchangeably, but in some instances finer distinctions or
nuances are required. In this book, we shall observe the
following definitions.
Numbers describe magnitude or position.
Cardinal numbers - zero, one, two, three ... fifty-six... - allow
us to count the objects or ideas in a given collection.
Ordinal numbers - first, second, third ... fifty-sixth... - state
the position of individual objects in a sequence.
Integers is the term used to denote zero and positive and
negative whole numbers: ... -3, -2, -1, 0, 1, 2, 3
Fractions are numbers that represent a part, or several equal
parts, of a whole: one-half, two-thirds, three-fifths, etc.
Numerals are symbols, or combinations of symbols, which
describe numbers.
Digits are specific symbols used - alone or in combinations -
to denote numbers.

Chapter 1 NUMBERS AND LANGUAGE
Arabic numerals are written with so-called Arabic digits,
alone,
0,1, 2, 3, 4, 5, 6, 7, 8, 9,
or in combinations,
10, 11,12 ... 1995 ...,
which are modifications of the original Hindu-Arabic number
signs.
Roman numerals are written with certain letters - I, V, X, L,
C, D, M - of the Latin alphabet,
I, II, III, IV, V, VI, VII, VIII, IX, X, XI, XII ... MCMXCV ....
Number Mystique
Many numbers have mystic or mythical associations: All
things good and bad are said to be three; five is found in the
pentagram of mystics and soothsayers, and in the pentacle
of Solomon; the seven arms of the Hebrew candelabrum,
and the Seven Wonders of the World (not always the same but
always seven in number); the proverbial nine lives of a cat,
and the cat-o'-nine-tails (though that is another kind of cat);
the belief that the number thirteen forebodes bad luck; etc.
A more realistic, even mathematical, association attaches to
the number forty, indicating "many" or "too many" with a
touch of annoyance or boredom: We remember Ali Bab a and
his forty thieves; Moses was away from his people for forty
days and forty nights when he was given the Ten
Commandments on Mount Sinai; the Children of Israel were foot-
slogging it for forty years through the wilds of the Sinai desert.
In many countries in the Near East a similar meaning
attaches to the number sixty, and the "1001 Nights" signifies a
kind of "finite infinity".
Arcane coincidence is attached to and between numbers, but:
"... With numbers you can do anything you lif^e. Suppose I have the
sacred number 9 and I want to get a number 1314, date of the execution
of Jacques de Mo (ay- a date dear to anyone who, life me, professes
devotion to the (Templar tradition of Iqiighthood. What do I do? I
multiply nine by one hundred and forty-si?Q the fateful day of destruction
of Carthage. How did I arrive at this? I divided thirteen hundred and
fourteen by two, three, et cetera, until I found a satisfying date. I could
also have divided thirteen hundred and fourteen by G.28, the double of
3.14, and I would have got two hundred and nine. That is the year
Attains I, Iqng ofTergamon, ascended the throne, you see?"
Umberto Eco, Foucault's Pendulum (1989)

1.2 Number Names
BACON Roger
(English philosopher and
scientist; introduced the
concept of the "laws of nature";
c. 1220 -1292)
... 'Bacon was right in saying that the conquest of learning is achieved
through the fqiowledge of languages,
Umberto Eco, The Name of the Rose (1983)
In this section we shall see how numbers are expressed in
various languages and how names for numbers developed in
English and some other languages. We will also discuss a
few items in the history of humankind's ability to count
things, among them the days and the months of the year.
Greenlandic
It is hard to imagine a language surpassing Greenlandic in
its practice of linking names of body parts with names of
numbers. Greenlanders use a twenty-base (vigesimal)
system of numeration, divided into recurring five-base
(quinary) periods of reckoning referring to the fingers and
toes of a person.
We begin by counting the fingers on one hand,
1 atuseq
2 mardluk
3 pingasut
4 sisamat
5 tatdlimat
and continue on the other hand,
6 arfineq
7 arfineq-mardluk
8 arfineq-pingasut
9 arfineq-sisamat
10 arfineq-tatdlimat
second hand
second-hand, two
second-hand, three
second-hand, four
second-hand, five
Ten is also called qulit.
We now go on to the toes of the first foot,
11
12
13
14
15
ind
16
17
18
19
20
arkaneq
arkaneq-mardluk
arkaneq-pingasut
arkaneq-sisamat
arkaneq-tatdlimat
first foot
first-foot, two
first-foot, three
first-foot, four
first-foot, five
then the toes of the second foot,
arfersaneq
arfersaneq-mardluk
arfersaneq-pingasut
arfersaneq-sisamat
arfersaneq-tatdlimat
second foot
second-foot, two
second-foot, three
second-foot, four
second-foot, five

Chapter 1 NUMBERS AND LANGUAGE
Twenty is also known as inuk n&vdlugo, which means "man
counted out". So you go on to the next person ... and the next ...
and the next...
32 inup aipagssane arqaneq-mardluk 2nd person, 12
48 inup pingajugssane arfineq-pingasut 3rd person, 8
78 inup sisamagssane arfersanek-pingasut 4th person, 18
94 inup tatdlimagssane arqaneq-sisamat 5th person, 14
... and so on.
For 100 and 1000, Greenlanders have created words based on
Danish names for these numbers,
100 = untrite Danish: hundrede
1000 = tusinte tusinde
106 = miliune million
1012 = piliune billion
Greenlanders also often turn to the Danish numbering system
to avoid the longer and tongue-twisting Iniipik expressions for
numerals higher than twenty, although this practice is
considered not quite polite in good company.
Purists will use the proper Iniipik words, so when you
are telling your friend Uvdluriaq about the uvak (codfish)
- sorry, uvarssuaq (big codfish) - that you caught in
arfersaneq-sisamanik untritigdlit arfiniq-pingasunigdlo
quligdit sisamatdlo
- that is, 1989 - be sure to say that it measured
inup tatdlimagssane arfersaneq-pingasut
- 98 centimeters, between the eyes, that is!

Section 1.2 Number Names
Maori Counting
North
Island J ^7
South
Island
Pacific Ocean
O^ew Zealand
New Zealand was first settled by a Polynesian people who
settled mainly on the North Island. These inhabitants lived
isolated for a long time and originally had no name for
themselves; later they adopted the name Maori - meaning normal
- to distinguish themselves from the European settlers.
The Maori used a decimal system of notation and generally
could count correctly to 180, but were a bit vague about higher
numbers - the word rau means 100, or "many"; mano is 1000,
or "very many".
There are traces of vigesimal counting in the system:
o hokotoru makete means three-score and maybe a little
more - that is, sixty-odd;
o hokowhitu is seven-score, that is, 140 - exactly or just about.
When counting things, cardinal numbers are preceded by the
word ka; when counting persons, prefixed by toko. Ordinal
numbers are preceded by tua.
one
two
three
four
Cardinal numbers when used
singly
tahi
rua
toru
wha
for things
ka tahi
ka rua
ka toru
ka wha
for persons
(to)kotahi
tokorua
tokotoru
tokowha
Ordinal
numbers
tua tahi
tua rua
tua toru
tua wha
Many things are counted in pairs:
ko tahi pu means a pair, a brace, or just two
ko rua pu, tautahi means two brace and one, that is, five
Today, virtually all Maori speak English; only a minority of
them also speak the Maori language.
Indo-European Languages
Protolanguage: an assumed or
recorded ancestral language
Merriam-Webster™
The majority of languages spoken today in Europe and the
Americas belong to the Indo-European family of languages.
The homeland was long believed to be on the Indian
subcontinent, but research makes that unlikely. A modern
hypothesis places the origin of Indo-European languages to
southeastern Europe in the area between the Black and Caspian Seas
- the Caucasus. The development from a common source could
have begun some 10 000 years ago, but the indication of time is
as controversial as the place of origin. Proto-Indo-European
might, in turn, have been a sister language of an even more
ancient family of languages.
The oldest language of the Indie branch of the Indo-European
family is Sanskrit, which was spoken until the 12th century
and is still used as a literary language. Being the language of
the sacred books of Hinduism and Buddhism, Sanskrit has
remained virtually unchanged over the past 3000 years and is
therefore of great importance to linguists studying the history
and interrelations of Indo-European languages.

Chapter 1 NUMBERS AND LANGUAGE
Number names are an important means in the research of the
relationship between languages.
Most of the European languages that are descended from
Proto-Indo-European belong to one of the following subgroups:
- Greek languages, in Southeastern Europe;
- Italic languages, in Southern Europe;
- Celtic languages, in Central Europe and in the British
Isles;
- Slavic languages, in Eastern Europe;
- Germanic languages, in Northern and Western Europe.
Greek
Classical Greek and Latin occur in a wealth of scientific
expressions, brought to us from the times when these
languages were used in communications between learned
people in the West. Today, the adoption of words from
Classical Greek and Latin into scientific expressions is
justified by the fact that the meaning of words in a "dead
language" is less apt to change, permitting the formation of
scientific terms that allow an exact and stable interpretation.
Greek has a longer documented history than any other Indo-
European language - from Ancient Greek in the 14th century
B.C. to Modern Greek today. Classical Greek dates from the
5th and 4th centuries B.C.
Greek number names meet us in a myriad of current English
words:
monograph pentagram ennead
diphthong hexameter decathlon
triglyph heptane hendecagon
tetralogy octopus the Dodecanese
... hecatomb ... kilogram ...
In fact, myriad is also of Greek origin, possibly stemming
from murmex - Greek for "ant" - suggesting abundance; the
word murmex later acquired the meaning of "countless", and
finally mitrioi came to mean 10000.
Alphabets
The Phoenicians, who lived in what is today roughly Lebanon,
were famous as tradesmen and for their skill as seafarers,
but their most remarkable contribution was their phonetic
alphabet, and with it the idea of sounding out words
phonetically. This was developed between c. 1700 and c. 1500 B.C.,
and was a model and forerunner of the Hebrew and Greek
alphabets.

Section 1.2 Number Names
The Greeks became acquainted with the Phoenician alphabet
around the 10th century B.C. and adapted it to suit their own
language.
The Greek Alphabet The Greeks modified some of the Phoenician characters and
added more letters to represent sounds in Greek not given in
the earlier symbols. The Greek alphabet is still essentially a
direct adoption of the ancient Phoenician alphabet, with the
basic phonetic values of the original characters retained.
The Greeks also embraced the names of the Phoenician
characters. For instance, the name of the letter alpha
- originally shaped like an ox head - can be traced back
to the Phoenician word for ox (aleph), chosen to represent
the "a" sound; beta goes back to the word for house (beth),
representing the "b" sound.
Typeset Handwritten
Upright Sloping
A
B
r
A
E
z
H
0
I
K
A
M
N
M
o
n
p
* At the end of a word £
T
T
<D
X
H*
Q
Upright letters are used in science and technology as symbols
for units of measurement (e.g., Q. for ohm), and sloping letters
(italics) as symbols for variable quantities (e.g., A for
wavelength).
To distinguish between upright and sloping handwritten
letters is hardly possible unless the writer indicates that, for
instance, underlined letters signify sloping letters (italics).
a
P
Y
6
e, €
C
Tl
d, e
i
K
X
n
V
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0
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P
a, g*
T
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X
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CO
A
B
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A
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P
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P
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d
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s
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K
P
e,g*
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V
<P> 0
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alpha
beta
gamma
delta
epsilon
zeta
eta
theta
iota
kappa
lambda
mu
nu
xi
omicron
Pi
rho
sigma
tau
upsilon
phi
chi
psi
omega
A
B
r
A
E
Z
H
e
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A
M
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P
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or
A
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Chapter 1 NUMBERS AND LANGUAGE
Transliteration of Greek Character
Greek characters are transliterated into Latin in the following
manner:
Greek
alpha
beta
gamma
delta
epsilon
zeta
eta
theta
iota
kappa
lambda
mu
A
B
r
A
E
Z
H
e
i,
K
A
M
a
P
Y
5
£
e
11
e
i
K
X
n
Latin
A
B
G
D
E
Z
E
Th
I
K
L
M
a
b
S
d
e
z
e
th
i
k
1
m
Greek
nu
xi
omicron
Pi
rho
sigma
tau
upsilon
phi
chi
psi
omega
N
**
O
n
p
2
T
T
O
X
*F
Q
V
%
0
71
P
a q*
T
V
<P
X
V
00
Latin
N n
X x
O o
P p
R r
S s
T t
U u (orY,y)
F f
Ch, ch (or H, h)
Ps ps
0 o
In Classical Greek, a vowel at the beginning of a word has
either a c or a ' over it, indicating a rough and a smooth aspirate
sound, respectively. In transliteration, only the rough aspirate
is denoted, either by c in front of the vowel or by beginning the
word with an "h"; thus, c£titoc (seven) can be written either cepta
or, as in this book, hepta.
Greek Heritage
Notable descendants of the Greek alphabet were the Etruscan
alphabet and the Cyrillic alphabet.
The Etruscan alphabet was modeled on the Greek alphabet in
the 7th century B.C. by the Etruscans, who enjoyed the most
highly developed civilization of all peoples in the Italian
peninsula before the rise of Rome. The Etruscan alphabet,
in turn, developed into the Latin alphabet, which is today
employed as the script of most Western languages.
The original Cyrillic alphabet was invented in the 9 th century
by the Greek missionary St. Cyril. For his translation of Holy
Writ into what became known as Old Church Slavonic, St.
Cyril came to be referred to as one of the two "Apostles of the
Slavs", the other being his elder brother St. Methodos.
The original Cyrillic alphabet had 43 characters; modern
Slavic languages have 30 - 34 letters.

Section 1.2 Number Names
13
Names of Numbers in Classical Greek
Cardinal Numbers Ordinal Numbers
Numeral Adverbs
* As in -•
f * f yf
hapax Legomenon, 2
a word or form 3
occurring only once 4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
30
40
50
60
70
80
90
100
200
300
1000
2000
10 000
heis, mia, hen
duo
treis, tria
tessares, tessara
pente
hex
hepta
okto
ennea
deka
hendeka
dddeka
treiskaideka
tessareskaideka
pentekaideka
hekkaideka
heptakaideka
oktdkaideka
enneakaideka
eikosi
triakonta
tessarakonta
pentekonta
hexekonta
hebdomekonta
ogdoekonta
enenekonta
hekaton
diakosioi, -ai, a
triakosioi, -ai, -a
chllioi, -ai, -a
dischllioi, -ai, -a
murioi, -ai, -a
protos
deuteros
tritos
tetartos
pemptos
hektos
hebdomos
ogdoos
enatos
dekatos
hendekatos
dodekatos
triskaidekatos
tessarakaidekatos
pentekaidekatos
hekkaideka tos
heptakaidekatos
oktokaidekatos
enneakaidekatos
eikostos
triakostos
tessarakostos
pentekostos
hexekostos
hebdomekostos
ogdoekostos
enenekostos
hekatostos
diakosiostos
triakosiostos
chfliostos
dischfliostos
mu riostos
hapax * once, 1 time
dis twice, 2 times
tris thrice, 3 times
tetrakis
pentakis
hexakis
heptakis
oktakis
enakis
dekakis
hendekakis
dodekakis
triskaidekakis
tessarakaidekakis
pentakaidekakis
hekkaidekakis
heptakaidekakis
oktokaidekakis
enneakaidekakis
eikosakis
triakontakis
tessarakontakis
pentekontakis
hexekontakis
hebdomekontakis
ogdoekontakis
enenekontakis
hekatontakis
diakosiakis
triakosakis
chiliakis
dischiliakis
muriakis
4 times
5 times
6 times
7 times
8 times
9 times
10 times
11 times
12 times
13 times
14 times
15 times
16 times
17 times
18 times
19 times
20 times
30 times
40 times
50 times
60 times
70 times
80 times
90 times
100 times
200 times
300 times
1000 times
2000 times
10 000 times
More than one form indicates inflection by gender - masculine, feminine, neuter.
The number noun monos, English monad, meaning "single",
"alone" is a common prefix signifying uniqueness or
seclusion - monandry, monarch, monastery.
Greek mathematicians in classical times had no symbol for
zero, nor any use for it, as they had not a full concept of its
mathematical significance, but words denoting nothingness
certainly existed:
oudeis, oudemia, ouden no one, nothing;
the three forms indicate inflection by gender - masculine,
feminine, and neuter, respectively.
Many Greek words were taken over by the Romans and
Latinized and have come down to us in these forms, which are
the forms often used in current scientific terminology.
Names of numbers in Modern Greek and Classical Greek
differ little from one another.

Chapter 1 NUMBERS AND LANGUAGE
Latin
What is thy name? And he answered, saying, My name is Legion:
for we are many.
Mark 5: 9
As with Greek, the imprints of Latin number names on the
English language are legion:
unicorn
biceps
tricycle
quartet
quintessence
sextant
Septuagint
octave
nonagenarian
decade
centennial
millipede
Legion itself is also of Latin origin, from legio which means
"army", suggesting untold numbers.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
30
40
50
60
70
80
90
100
101
200
300
400
500
600
700
800
900
1000
2000
10 000
105
106
Cardinal Numbers
unus, una, unum
duo, duae, duo
tres, tres, tria
quattuor
quinque
sex
septem
octo
novem
decern
undecim
duodecim
tredecim
quattuordecim
quindecim
sedecim
septendecim
duodeviginti
undeviginti
viginti
viginti unus
triginta
quadraginta
quinquaginta
sexaginta
septuaginta
octoginta
nonaginta
centum
centum unus
ducenti
trecenti
quadringenti
quingenti
sescenti
septingenti
octingenti
nongenti
mille
duo milia
decern milia
centum milia
decies centena milia
Ordinal Numbers
primus, first
secundus, alter, second
tertius
quartus
quintus
sextus
Septimus
octavus
nonusl4
decimus
undecimus
duodecimus
tertius decimus
quartus decimus
quintus decimus
sextus decimus
Septimus decimus
duodevicesimus
undevicesimus
vicesimus
vicesimus primus
tricesimus
quadragesimus
quinquagesimus
sexagesimus
septuagesimus
octogesimus
nonagesimus
centesimus
centesimus primus
ducentesimus
trecentesimus
quadringentesimus
quingentesimus
sescentesimus
septingentesimus
octingentesimus
nongentesimus
millesimus
bis millesimus
decimus millesimus
centies millesimus
decies centies millesimus
Numeral Adverbs
semel, once
bis, twice
ter, thrice
quater
quinquies
sexies
septies
octies
novies
decies
undecies
duodecies
ter decies
quater decies
quinquies decies
sexies decies
septies decies
duodevicies
undevicies
vicies
semel et vicies
tricies
quadragies
quinquagies
sexagies
septuagies
octogies
nonagies
centies
centies semel
ducenties
trecenties
quadringenties
quingenties
sescenties
septingenties
octingenties
noningenties
milies
bis milies
decies milies
centies milies
decies centies milies
Distributives
singuli, one by one
bini, two by two
terni or trini,
quaterni
quini
seni
septeni
octoni
noveni
deni
undeni
duodeni
terni deni
quaterni deni
quini deni
seni deni
septeni deni
duodeviceni
undeviceni
viceni
viceni singuli
triceni
quadrageni
quinquageni
sexageni
septuageni
octogeni
nonageni
centeni
centeni singuli
duceni
treceni
quadringeni
quingeni
sesceni
septingeni
octingeni
nongeni
singula milie
bina milia
deni milia
centena milia
decies centena milia

Section 1.2 Number Names
15
Etruscan Language
millio
Greek:
heis, duo, treis -
tessares, pente, hex -
hepta, okto, ennea -
deka-
More than one form indicates inflection by gender
- masculine, feminine, neuter. Cardinal hundreds 200 - 900
are inflected by gender: -i, -ae, -a. All ordinal numbers are
inflected by gender and number: -us, -a, -um; -i, -ae, -a.
Observe that
18 : duodeviginti
19 : undeviginti
analogously,
28 : duodetriginta
29 : undetriginta
means
means
means
means
two-from-twenty,
one-from-twenty;
two-from-thirty,
one-from-thirty, etc.
These forms were inherited from the Etruscan language,
which was spoken in Etruria, the ancient country of western
central Italy, now in Tuscany and part of Umbria:
17
18
19
27
28
29
ci-em zathrum
esl-em zathrum
thun-em zathrum
ci-em-ce-alch
esl-em-ce-alch
thun-em-ce-alch
three-from-twenty
two-from-twenty
one-from-twenty
three-from-thirty
two-from-thirty
one-from-thirty
In Medieval Latin, the word millio appeared as an
augmentative form of mille, meaning "great thousand", that is to say a
million (= 106).
Like the Greeks, Romans had neither a symbol for zero nor a
clear concept of its mathematical significance, but words
denoting nothingness existed:
nullus, -a, -um
nulli (plural)
nil, nihil
no one
nothing
A comparison of the Greek and Latin names for lower
numbers reveals several similarities pointing to early Greek
influence. The schoolmaster scans:
Unus, duo, tres -
tu adire scholam debes.
Quattuor, quinque, sex -
ibi non es rex.
Septem, octo, novem -
te faciunt officiosum civem.
Decern -
jam comprendis legem.
Unus, duo, tres -
you ought to go to school.
Quattuor, quinque, sex -
there you are no ruler.
Septem, octo, novem -
you be made a civil servant.
Decern —
now you know the terms.

Chapter 1 NUMBERS AND LANGUAGE
To express fractions, Latin has words for subunits of twelve:
1/6
1/4
1/3
1/2
2/3
3/4
5/6
1
1/12
2/12
3/12
4/12
5/12
6/12
7/12
8/12
9/12
10/12
11/12
12/12
uncia
sextans
quadrans
triens
quincunx
semis
septunx
bes
dodrans
dextans
deunx
as
meaning
one-twelfth of unity
one-sixth
one-quarter
one-third
five-twelfths
one-half
seven-twelfths
two-thirds
three-fourths
five-sixths
eleven-twelfths
one, the whole, unitv
based on
unus, one
sex, six
quattuor, four
tres, three
quinque, five + uncia
semi- (half-) + as
septem, seven + uncia
bis-assi-s;
two (portions) short
de + quadrans;
one quarter short
de + sextans;
one sixth short
de + uncia;
one twelfth short
There are also subdivisions of the uncia:
semuncia one-half semis, one half
duella
sicilicus
sextula
one-third diminutive of duo, two (sixths);
also called binae sextulae
one-quarter from secula, crescent, sector
one-sixth diminutive of sextus
scrupulus: a small chip of a stone
scripulum 1/24
Roman measurements for weight and mass were the uncia
and the libra:
1 uncia = 27 -27.5 g
12 unciae = 1 libra
Roman copper as (Janus face)
The as - the whole, unity - was Rome's basic unit of money,
originally worth one libra of copper, later depreciated to one-
half uncia.

Section 1.2 Number Names
17
HORACE
(65 - 8 B.C.)
Mastery of the monetary system was of great importance
in Roman education, as recounted by Horace (65 - 8 B.C.) in his
Ars poetica (lines 325 - 330):
Romani pueri longis rationibus assem
discunt in partis centum diducere.
"dicat filius Albini: si de
quincunce remota est uncia, quid superat?
poteras dixisse."
"triens."
"eu! rem poteris servare tuam.
redit uncia, quid fit?"
semis.
p. 39
p. 168
Through lengthy reckonings, boys in Rome
learn to divide the as into hundredths.
"Tell me, son of Albinus: if from
a quincunx, an uncia is taken, what is left?
Please, say."
"A triens."
"Good! You'll be able to manage your assets.
An uncia added; what does it make?"
"A semis."
Latin uncia gave both "inch" (Old English: ynce; Middle
English: inche) and "ounce" (Middle English: unce).
The English abbreviation lb, the £ sign, and the word "pound"
all have Latin forebears: £ and lb = libra; pondo means "in
weight"; its Latin abbreviation P is for {librae) pondo.
The Roman notations of numeration will be described in
Chapter 2, Roman use of the abacus in Chapter 4.
Mathematics in Ancient Rome
WiOtk
Tolle matHematicum ! est augur surdus, haruspe^ cecus et hariolus
demens: praesentia scire Jos Hominum est, soli 'Domino praescire
futura
From Hans Walther (Gottingen, 1963): Proverbia Sententioeque Lotinitotis
"Take the mathematician away: He is a stupid augur, blind
prophet, a crazy soothsayer. Man may know the present;
only God can foresee the future."
In contrast to the high regard in which mathematics and
mathematicians were held in ancient Greece, Romans
considered mathematics an unworthy subject for a noble Roman
to pursue. Being more concerned with soldiering and
politicking, only simple reckoning found favor with them.
Of old, the Romans used a ten-month calendar of 304 days,
followed by a winter whose days nobody seems to have kept
track of. The year began with March. The days of winter were
later accounted for in the new months of January (at the end of
the year!) and February (at the beginning of the year!),
allegedly introduced by King Numa in 717 B.C. - eventually
bringing the length of the calendar year up to 355, or 356, days,
with every other year a leap year.
According to modern study, the calendar of Old Rome might
not have been as bad as first thought; the description might
have been spurious propaganda to enforce the introduction of a
new calendar.

Chapter 1 NUMBERS AND LANGUAGE
PRISCUS Tarquinius
(616 - 579 B.C.)
CAESAR Gaius Julius
(c. 100 - 44 B.C.)
SOSIGENES
(1st century B.C.)
About 600 B.C., the Etruscan Tarquinius Priscus introduced the
Roman Republican Calendar, a lunar calendar of 355 days,
that is, 10 and 1/4 days short of the solar year. To compensate
for this deficit, the Romans wedged in an extra month,
Intercalarius, after 23 February, conveniently eliminating
the remaining 5 days of February; in this manner, a calendar
year averaging 366 and 1/4 days in a four-year period was
achieved. This is mathematics after the Roman fashion.
The decision when and how to introduce the Intercalarius was
the responsibility of a Roman Board of Magistrates, whose
deliberations were theoretically secret, but who were not averse
to receiving bribes, if ample enough - also a kind of Roman
mathematics - causing the calendar to proceed by leaps and
rebounds.
In 46 B.C., Julius Caesar realized that drastic measures were
required and called upon Sosigenes of Alexandria, a
reputable Greek astronomer and mathematician, to bring order out
of chaos. He immediately decided that the only possible
remedy was to abandon the lunar calendar altogether, and he
corrected the accumulated errors by inserting no fewer than
three extra months in the year 46 B.C. Then, he introduced a
365-day year with an extra day placed between the 23rd and
24th of February every four years - the punctum temp or is.
This calendar came to be known as the Julian calendar.
The new calendar proved too much for the Romans and got out
of hand; the last year of a four-year period was mistakenly
taken as the first year of the following period. This went
undetected for 36 years until, in 8 B.C., the Emperor Augustus
ordered the canceling of three leap days, and in A.D. 4 the
Julian calendar finally came abreast of time.
GREGORIUS XIII
(1502 -1585)
The Gregorian Calendar
The Julian calendar contained a seemingly insignificant
error of 11 minutes and 14 seconds annually, but as the
years passed, this shortfall continued steadily to build up into
days. When Ugo Buoncompagni, a former teacher of law
at the University of Bologna, became Pope Gregory XIII in
1572, he inherited the problem of a calendar that was 10 days
short; the vernal equinox fell on 11 March instead of its proper
date of 21 March.
On the 24th of February 1582, acting on the advice of two
renowned astronomers, Gregory XIII issued a papal edict
directing that - to allow the calendar to catch up with the Lord's
Time - the accumulated error of 10 days be removed by letting
the day following the 4th of October 1582 be the 15th of October.
The Gregorian calendar has an extra day (29 February) every
fourth year, except - and that is what distinguishes it from the
Julian calendar - in centennial years unless the year is
exactly divisible by 400; so, e.g., the year 1600 was a leap year,
but not 1700,1800, and 1900.
The Gregorian calendar, which became known as the New
Style, was recognized immediately by nearly all countries

Section 1.2 Number Names
19
of Catholic faith. Protestant countries were more resistant,
with Protestant German states adopting the calendar in 1700,
Britain in 1752, and Sweden in 1753. Countries of the Greek
Orthodox persuasion in Eastern Europe adopted it in 1912 - 17,
Greece in 1923, and atheist U.S.S.R. in 1918.
Monthly Number-Quandary
The misleading names of our ninth, tenth, eleventh, and
p. 17 twelfth months are derived from the ten-month calendar
- which started with March (Martius) - where September,
October, November, and December were the seventh, eighth,
cf. Latin number names,p. 14 ninth, and tenth months. There were also Quinctilis and
Sextilis, later to became July (Julius; after Julius Caesar) and
August (Augustus; after the first Roman emperor).
Modifications of Latin
The oldest evidence of Latin is an inscription in Greek
characters dating from the 6th century B.C. Spoken originally
by small groups of people living along the banks of the Tiber,
Latin became the standard language of most of the Roman
Empire, spreading throughout Southern and Western Europe
and into large parts of the coastal regions of North Africa.
In A.D. 395 the Roman Empire split in two. The Western
Empire fell in 476, the Eastern in 1453. Latin as a spoken
language became extinct in the period of the 7th to 10th
centuries, although it remained the official language of the
Church. It also served as a language of general
understanding among scientists, and it held a place as the language
of diplomatic service well into the 18 th century, when it was
ousted by French.
Written Latin distanced itself from its models in form and
substance until the 15 th century, but the reawakening of an
interest in science during the Renaissance saw a determined
return to the Latin of the Golden Age (81 B.C. - A.D. 14 ).
European scholars continued to use Latin extensively as a
written language well into the first half of the 19th century.
GAUSS Carl Friedrich Carl Friedrich Gauss, the eminent German mathematician,
(1777-1855) published all his major works in Latin, and at Oxford all
university business was transacted in Latin until 1854.
Fraktur
Fraktur is a style of letters formerly used in German
manuscripts and printing. Capital letters of boldface Fraktur, e.g.,
H (A), ¢(0), $(D), 5(F), $(P), ©(S),
are sometimes used in mathematics for sets of numbers.

20 Chapter 1 NUMBERS AND LANGUAGE
Romance Languages
The Romance languages - the major ones being Italian,
French, Spanish, Portuguese, and Romanian - are
descendants of Latin.
As the Roman Empire grew, soldiers, administrators,
tradesmen, and settlers were sent into the conquered provinces,
introducing not only Greek culture and Roman law, but also
colloquial Latin, which continued to develop on its own long
after Roman rule came to an end, thus giving rise to the
languages we now refer to as Romance languages. As Latin is
preserved in writing, Latin and the Romance languages
constitute an unusually solid basis for the study of the
relationships between the mother language and its descendents.
Latin, Romanus, "Roman" The words "romance" and "romantic" did not originally
connote love or (short-lived) enthusiasm. Romance was the
Latin of everyday life, as distinguished from book Latin.
The common heritage of the Romance languages is evident
when the names for numbers are compared. Spanish and
Portuguese are linguistically very close; of the two, we choose,
arbitrarily, Spanish.
Italian:
c before i and e pronounced like English ch
French:
Celtic influence on French number names
will be discussed on p. 24
* Archaic and provincial forms:
septante = 70
huitante = 80
nonante = 90
Spanish:
t y means and
1
2
3
4
5
6
7
8
9
10
11
12
13
c and z before i and e in Spain pronounced as ^4
in English thing, in Central and South -i c
America as in sing
lb
c before a,o,u pronounced like k in English .. n
hind' oh lilcp Encnifln ch
KlXXyXy U/I» 1XX^.C IZJkLfZLltjll 1^/1»
Romanian:
c before i and e pronounced like English ch;
z is like an English voiced s
3 as in English sh
** spre: above, higher
18
19
20
30
40
50
60
70
80
90
100
1000
2000
Italian
uno, una
due
tre
quattro
cinque
sei
sette
otto
nove
dieci
undici
dodici
tredici
quattordici
quindici
sedici
diciassette
diciotto
diciannove
venti
trenta
quaranta
cinquanta
sessanta
settanta
ottanta
novanta
cento
mille
duemila
French
un, une
deux
trois
quatre
cinq
six
sept
huit
neuf
dix
onze
douze
treize
quatorze
quinze
seize
dix-sept
dix-huit
dix-neuf
vingt
trente
quarante
cinquante
soixante
soixante-dix*
quatre-vingts*
quatre-vingt-dix*
cent
mille
deux mille
Spanish
uno, una
dos
tres
cuatro
cinco
seis
siete
ocho
nueve
diez
once
doce
trece
catorce
quince
diez y seis -f-
diez y siete
diez y ocho
diez y nueve
veinte
treinta
cuarenta
cincuenta
sesenta
setenta
ochenta
noventa
cien
mil
dos mil
Romanian
un, una
doi, dou&
trei
patru
cinci
§ase
§apte
opt
nou&
zece
unsprezece**
doisprezece
treisprezece
patrusprezece
cincisprezece
§asesprezece
§aptesprezece
optsprezece
nou&sprezece
dou&zeci
treizeci
patruzeci
cinci zeci
§asezecf
§aptezeci
optzeci
nou&zeci
osut&
o mie
dou&mii

Section 1.2 Number Names
21
Rhaeto-Romance Languages
(1st century)
The ancient country of Rhaetia - comprising Vorarlberg and
Tirol in Austria, parts of Northern Italy, the Eastern cantons
of Switzerland, and parts of Bavaria and Baden-Wiirttemberg
in Germany - was invaded and made a province under
Roman rule in the year 15 B.C., starting a Latinization of the
native language. The Latinization was followed by German
influence from the 6th century onward.
There are two main branches of the Rhaeto-Romance
languages, Romansh and Ladin, with five dialects between
them. The kinship with other Romance languages is
demonstrated by comparison with the Italian names for the
numbers 1-10 and 100.
1
2
3
4
5
6
7
8
9
10
100
Italian
uno
due
tre
quattro
cinque
sei
sette
otto
nove
dieci
cento
Romansh
in, egn
dus
tres, treis
quatar, quater, catter
tschun, tschentg,
sis, seis
siat, seat, set
otg
nov, nof
diesch, diasch
tschintg
tschien, tsciant, tschient
Ladin
A
un
duos
trais
quatter
tschinch
ses
set
och, ot
nouv
desch
tschient
Celtic Languages
LINCOLN Abraham
(1809-1865)
16th U.S. President
Jour score and seven years ago our fathers brought forth on this
continent a new nation ...
Abraham Lincoln: Gettysburg Address (1863)
Four score and seven is not an everyday expression in the
English language. President Lincoln no doubt chose it
wittingly because of its old-time flavor in an effort to create
among his listeners an atmosphere of solemnity befitting the
occasion.
The olden practice of counting in terms of twenties was in
general use in most English-speaking countries as late as a
few generations ago, and is still the rule in many existing
languages.
Over a period from approximately 500 B.C. to A.D. 500, Celtic
languages were spoken in a large area comprised by western,

Chapter 1 NUMBERS AND LANGUAGE
central, and southern Europe. Celtic conquerors and
immigrants from Gaul (Gallia; in essence modern Belgium and
France) spread to areas corresponding to modern northeastern
central Spain and the northern part of Italy. Celts also
penetrated into Asia Minor and created great destruction
among Hellenistic states. At the beginning of the 3rd century
B.C., Celts became settled in a region to be known as Galatia
- centered on modern Ankara, Turkey; its name alluded to
the Galatae, or Galatians, the appellation used for Celts by
writers of that time. By the 2nd century A.D. the Galatians had
become absorbed into the Hellenistic civilization. The Celtic
languages in Europe were gradually repressed by Germanic
and Roman languages and are presently spoken only in
limited territories of the British Isles and in Brittany
(Bretagne) where, in A.D. 5th or 6th century, a Celtic language
appeared as a result of colonization from Britain.
Vigesimal counting is a feature of both the remaining groups
of Celtic languages: the Goidelic, or Gaelic, group consisting
of Irish, Scots Gaelic, and the recently extinct Manx; and the
Britannic group comprising Welsh, long extinct Cornish, and
Breton, still spoken in parts of Brittany.
For English speakers there is nothing difficult - except
spelling and pronunciation - about the first ten numerals in
the Celtic languages,
Irish
Soots
Welsh
1
2
3
4
5
6
7
8
9
10
aon
do
tri
ceathair
cuig
se
seacht
ocht
naoi
deich
where double forms in
nine gender.
aon
da
tri
ceithir
coig
sia
seachd
ochd
naoi
deich
un
dau, dwy
tri, tair
pedwar, pedair
pump
chwech
saith
wyth
naw
deg
Welsh denote masculine versus femi-
Let's practice with some Welsh children's
rhymes:
Mae gen i
Un gaseg yn pori,
Dwy yn y plwy'
Tair yn y ffair,
Pedair heb ddim pedol,
Pump yn eu crimp
Chwech yn y frech,
Saith yn y gwaith
Wyth yn y ffrwyth
Naw yn y galw,
A deg yn pori gwair
Ton-teg.
I have
Un mare grazing,
Dwy in the parish,
Tair in the fair,
Pedair without a shoe,
Pump pinching them,
Chwech vaccinated,
Saith in the works,
Wyth in the effect,
Naw visiting,
And deg grazing the hay of
Ton-teg.

Section 1.2 Number Names
23
* * *
Un, dau, eto tri
Mi wala* ar y ty.
Pedwar, pump, eto chwech,
Dyna blentyn yn rhoi sgrech.
Saith, wyth, naw, deg,
Eto'n dilyn un ar ddeg.
Un, dau, also tri
I see on the house.
Pedwar, pump, also chwech,
Here is a screaming child.
Saith, wyth, naw, deg,
Also un ar ddeg following.
Nor is there anything unexpected in the numbers 11 to 20 in
Irish and Scots Gaelic, whereas Welsh has a more tortuous
way of expressing these numbers, particularly from 16 to 19:
11
12
13
14
15
16
17
18
19
20
Irish
aondeag
dodheag
tridheag
ceathairdeag
ciiigd£ag
sedeag
seachtdeag
ochtdeag
naoideag
fiche
Scots
aon deug
da dheug
tri dheug
ceithir dheug
coig dheug
sia deug
seachd deug
ochd deug
naoi deug
fiche ad
Welsh
un ar ddeg
deuddeg
tri ar ddeg
pedwar ar ddeg
pymtheg
un ar bymtheg
dau ar bymtheg
deunaw
pedwar ar bymtheg
ugain
meaning
one and ten
two-ten
three and ten
four and ten
five-ten
one and fifteen
two and fifteen
twice nine
four and fifteen
twenty
Irish
Scots
Welsh
* Welsh 70 is also:
deg a thrigain
** Welsh 90 is also:
deg a phedwar ugain
20
30
40
50
60
70
80
90
100
fiche
deich fichead
daichead
deich ar daichead
tri fichead
deich is tri fichid
cheithre fichid
deich is cheithre fichid
cead
fichead
deich air fichead
da fhichead
da fhichead
's a deich
tri fichead
tri fichead
's a deich
ceithir fichead
ceithir fichead
's a deich
ceud
ugain
deg ar hugain
deugain
deg a deugain
trigain
trigain a deg *
pedwar ugain
pedwar ugain
a deg **
cant
In addition to the older system, modern Welsh has a decimal
system:
11
12
20
30
undeg un
undeg day
daudeg
trideg
etc.
etc.
Welsh zero is dim, "nothing".

Chapter 1 NUMBERS AND LANGUAGE
Celtic Remnants in French and Danish
French still uses remnants of a vigesimal system of counting
inherited from its Celtic predecessor, the language of the Gaul.
70 soixante-dix
71 soixante-onze
72 soixante-douze
79 soixante-dix-neuf
80 quatre-vingts
90 quatre-vingt-dix
99 quatre-vingt-dix-neuf
sixty-ten
sixty-eleven
sixty-twelve
sixty-ten-nine
four-twenties
four-twenties-ten
four-twenties-and-nineteen
Vigesimal counting in Old French is evidenced by another
relic in the name of a Paris hospital built in the 13th century to
accommodate 300 blind people, which is still called the
Hopital des Quinze-Vingts,
that is, the Hospital of the Fifteen Score.
Danish, a North Germanic language, has a rather roundabout
way of expressing numbers from fifty onward, by counting
"halves", also a legacy from an otherwise forgotten vigesimal
system.
50
60
70
80
90
halvtreds
tres
halvfjerds
firs
halvfems
or, in full:
halvtredsindstyve
tresindstyve
halvfjerdsindstyve
firsindstyve
halvfemsindstyve
treds = third
fjerds = fourth
ferns = fifth
halv = half
sinds = times
tyve = twenty
All this becomes easier to understand if we realize that
l
2
9.1
is the first half
is the second half
is the third half
is the fourth half
is the fifth half
Danish: halvanden
halvtreds
halvfjerds
halvfems
So, clearly, halvfjerds is 3 — x 20 = 70

Section 1.2 Number Names
25
Germanic Languages
Protolanguage: an assumed or
recorded ancestral language
Merriam-Webster™
The original Germanic language - Proto-Germanic -
separated into three subfamilies:
- East Germanic, via extinct Gothic, the parent language
of Burgundic, Vandalic, Ostrogothic, and Visigothic;
- North Germanic, which turned into Old Norse from which
developed Icelandic, Faeroese, Danish, two Norwegian
languages, and Swedish;
- West Germanic, parent language of Old Saxon, Old High
German, Old Frisian, and Old English, all well attested.
Number Names of Some Modern Germanic Languages
Icelandic:
/> indicates a voiceless lisping sound,
as in think and three.
Swedish:
sj indicates a sound similar to the first
sound in shoe.
a indicates a sound similar to ou in
four, y is similar to German u.
Dutch:
-if is like English ay, as in. pay.
-ig is like ch in Scottish loch.
German:
u is a sound like u in bull.
ii has no equivalent English vowel
sound;
v is pronounced as f.
z is pronounced ts as in its.
s is a voiced s sound at the beginning
of a word, as in zeal.
chs is pronounced ks as in mix.
cht is a sound similar to Scottish ch in
loch plus the t sound.
■ig is a sound similar to the last sound
in catch.
Icelandic
0 nul
1 einn
2 tveir
3 {)rfr
4 fjorir
5 fimm
6 sex
7 sjo
8 atta
9 niu
10 tiu
11 ellefu
12 tolf
13 {jrettan
14 fjortan
15 fimmtan
16 sextan
17 sautjan
18 atjan
19 nitjan
20 tuttugu
21 tuttugu og einn
2 2 tuttugu og tveir
2 3 tuttugu og J)rir
Swedish
noil
ett
tva
tre
fyra
fern
sex
sju
atta
nio
tio
elva
tolv
tretton
fjorton
femton
sexton
sjutton
arton
nitton
tjugo
tjugoett
tjugotva
tjugotre
Dutch
nul
een
twee
drie
vier
vijf
zes
zeven
acht
negen
tien
elf
twaalf
dertien
veertien
vijftien
zestien
zeventien
achtien
negentien
twintig
eenentwintig
tweeentwintig
drieentwintig
German
null
eins
zwei
drei
vier
fiinf
sechs
sieben
acht
neun
zehn
elf
zwolf
dreizehn
vierzehn
funfzehn
sechzehn
siebzehn
achtzehn
neunzehn
zwanzig
einundzwanzig
zweiundzwanzig
dreiundzwanzig
30
40
50
60
70
{)rjatiu
fjortiu
fimmtiu
sextiu
sjotiu
trettio
fyrtio
femtio
sextio
sjuttio
dertig
veertig
vijftig
zestig
zeventig
dreissig
vierzig
fiinfzig
sechzig
siebzig

Chapter 1 NUMBERS AND LANGUAGE
Etymology of English Number Names
Old English came to England in the 5th and 6th centuries with
the Anglo-Saxons - a German people who settled in Britain
when Roman rule crumbled. From the late 8th century to well
into the 11th century Britain was ravaged and pillaged by
Norse Vikings whose language, Old Norse, was quite similar
to Old English.
An important event in the history of the English language was
William the Conqueror's victory at the Battle of Hastings
in 1066, which paved the way for the massive influx of French
words into English that was to continue for more than
500 years. Curiously enough, this linguistic offensive made
only scant inroads into well-established English names of
numbers.
Zero
Zero derives from Hindu sunya - meaning "void",
"emptiness" - via Arabic sifr, Latin cephirum, and Italian zevero;
sifr is also the origin of cipher.
One...Ten
Nearly all English names of numbers are of Germanic
origin, based on the words one, two, three, four, five, six, seven,
eight, nine, and ten, reflecting humankind's early use of
fingers to illustrate numbers, and a forerunner of both number
names and number symbols.
The word ten may actually have come from an old Indo-
European word meaning "two hands", transformed into
Gothic tai-hun, and into Old and Middle English tien.
1
2
3
4
5
6
7
8
9
10
Sanskrit *
eka
dva
tri
catur
panca
sas
sapta
asta
nava
daca
Old Norse *
einn
tveir
t>rir
fjorir
fimm
sex
sjau
atta
niu
tiu
Old
English * *
ane, an
twa
fcri
feower
fif
siex
seofan
eachta
nigon
tien
Middle
English * *
oon, on
twa, tu
t>ri
four(e)
fif
sex
seofan
eighte
nigon
tien
Modern
English
one
two
three
four
five
six
seven
eight
nine
ten
p indicates a voiceless lisping sound, as in think and three.
Diacritic symbols have been left out in transliteration.
* Other transliterations are possible.
* Several other forms existed.

Section 1.3 Etymology of English Number Names
27
One and the Indefinite Article
The Old English (c. 550 - c. 1100) word ane served both as a
counting number and as the indefinite article. Toward the end
of the Old English period and the beginning of Middle English
(c. 1100 - c. 1500), ane developed two pronunciations, the first
being used for the number 1 and the other for the indefinite
article an, a.
The existence of different words for the number one and the
indefinite article seems to be a unique feature of the English
language.
Eleven, Twelve (Dozen)
The names of numbers beyond ten are derived from
combinations of number words.
Eleven is a derivative of Old English endleofan, where the
terminal -fan is believed to carry the meaning of "to leave" or
"left", that is, one left after counting ten (fingers).
Twelve has developed in a similar manner from Old English
twelfe meaning two left over after counting ten; it may be
more clearly seen from the Gothic twailif.
Dozen is one of the few French loans; douze means twelve,
douzaine a dozen.
Thirteen... Twenty (Score)
Thirteen comes of three-and-ten, and the remaining teens
follow the same pattern.
Twenty was twentig or twegentig in Old and Middle English,
where twen and twegen were forms meaning two, and the
suffix -tig recalls German -zig.
Score originally meant a cut or scratch made on a tally-stick
of wood, a tablet of clay, or the like, in order to keep a record.
Twenty was marked by a longer scratch; hence the
development of a score to mean twenty.
Thirty... Ninety
All multiples of ten are formed in the same manner as twenty.
Hundred
Hundred derives from Old English hund, meaning a
hundred, allied to the Gothic taihun-taihund, meaning ten-
tens; the suffix -red is a later addition, a word for reckoning
or ratio - a hundred-number.
In the past, hundred also had the meaning of 120, which was the
original meaning of Old Norse hund-rath. English
differentiated between the hundred for 100 and the long hundred for
120.
Thousand
Thousand is akin to Old Norse thushund, giving the meaning
of great-hundred. The prefix denoting great is of the same
origin as in thumb, literally the stronger finger.

28
Chapter 1 NUMBERS AND LANGUAGE
Million
A thousand times a thousand is called a million, from the
Medieval Latin millio, which means great-thousand.
Beyond the Million
A thousand million is called a milliard in French, the word
being used also in other Romance languages and in
Germanic languages, though rarely in English.
Higher powers of ten, in steps of a million, are given names
compounded from a Greek number prefix plus million, thus,
(106)2 = bi-]
(106)3 = tri-
million =
million =
(106) = quadri-million = qua
The powers of ten will then be:
106 million
1012 billion
1018 trillion
1024 quadrillion
1030 quintillion
1036 sextillion
1042 septillion
1048 octillion
1054 nonillion
1060 decillion
1066 undecillion
1072 duodecillion
1078 tredecillion
billion
trillion
idrillion
10102
10108
10114
10120
■ ■ •
1084 quattuordecillion 10600
10 9 ° quindecillion
1096 sexdecillion
= 10^
= 1018
= 1024
septendecillion
octodecillion
novemdecillion
vigindecillion
centillion
The United States uses, unfortunately, another system - based
on an old French system - of naming numbers greater than a
million:
Non-U.S.
million = 106
= thousand million = 109
billion = 1012
quadrillion = thousand billion = 1015
etc.
In mathematics and science, this ambiguity is apt to cause
serious misunderstandings, and anybody of a scientific turn
of mind is well advised to be wary of the "zillions" and keep to
the purely metric numeric symbols of powers of ten.
1000-
1000-
1000-
1000
10001
10002
10003
10004
U.S.
= million
= billion
= trillion
= quadril
Global
Facje lifts
Move from the United States and your
one billion deficit mill be 1/1000 of a billion.
Let's go for it!
^^^L
Global
j-inanciQ-l
s
Move to the United States and your
thousand million profit will be a billion.
Let's go for it!

29
1.4 Numbers vs. Infinity
fAnd the angel of the Lord said unto her, I will multiply thy seed
e?(ceeding(y, that it shall not be numbered for multitude.
Genesis 16:10
ARCHIMEDES
(287?-212 B.C.)
KASNER Edward
(1876 -1955)
Archimedes is the first person known to have discussed
extremely large numbers. In his essay The Sand Reckoner, he
built a system on the myriad, the name for 10 000. He called
the numbers up to a myriad myriads (100 000 000) numbers of
the first order of the first period.
With 100 000 000 as the unit of the second order of the first
period, Archimedes could name numbers as large as
100 000 0002, and by continuing through the myriad-myriadth
order of the first period, he could extend the naming to
100 000 000100 °°° 00°.
Moving on to the second period (whose first order has
100 000 000100000000 for the unit), the third period, the fourth
period, and so on through the myriad-myriadth order of
the myriad-myriadth period, Archimedes had a system for
naming all numbers up to 10^0 000 000 000 000 000^ that is? a i
followed by 80 000 billion (billion = a million million) zeros.
The name googol was coined in the late 1930s by the nine-year-
old nephew of the American mathematician Edward Kasner
when he was asked to come up with a name for a very large
number; at the same time googolplex was suggested for a still
larger number.
Kasner gave definitions for these two number names:
1 googol = 10100,
that is, a one followed by 100 zeros, and
1 googolplex = 10S°°so1 = 10lol0°,
a one followed by a googol of zeros.
Infinity
In The Sand Reckoner, Archimedes demonstrated that the
number of sand grains the size of a poppy seed that could fill up
a volume equal to that of the then known Universe was fewer
than 1063, that is, 1 followed by 63 zeros.

30
Chapter 1 NUMBERS AND LANGUAGE
GAUSS Carl Friedrich
(1777 -1855)
Carl Friedrich Gauss called
p. 257
WALLIS John
(1616 -1703)
BERNOULLI Jakob
(1654-1705)
BERNOULLI Nikolaus
(1662 -1716)
oo
99"
eine messbare Unendlichkeit
that is, "a measurable infinity". The number has something
like 10369 ^3 10° digits, which makes it suitable for those who
would like to put a tag on every electron in the Universe.
In ordinary conversation, infinite means something that is
very great in comparison with everyday things. In
mathematics, however, infinity is not a number but a concept of
increase beyond bounds.
A collection like Archimedes* grains of sand, which can have
no more than a certain number of elements, is finite; here
number means an ordinary counting number: 1, 2, 3 ...
through the googol and googolplex, through the last number of
Archimedes* myriad-myriadth order of the myriad-myriadth
period ....
A collection such as that of all counting numbers, which
continues indefinitely, is infinite, and a process that may be
continued indefinitely is also infinite.
Although the concept of infinity is difficult to grasp, we may
simply define it as not finite, where finite means something
that - at least in theory - is completely determinable by
counting or measurement.
Infinity has calculation rules of its own:
00 + 00 = 00 *
00 • 00 = 00
Further aspects on infinity are discussed in section 7.5,
Transfinite Numbers.
The mathematical symbol of infinity was introduced in 1655
by John Wallis in his Arithmetica infinitorum, but did not
appear anew in print until the Ars conjectandi by Jakob
Bernoulli, published posthumously in 1713 by his nephew
Nikolaus Bernoulli (Nikolaus 1).
*««Krtt«eri|*fl
Treacher:
- "Eternity is infinitely long, indeed much longer
than that. Just thinf^of it, dear friends, that
when millions and millions of years have passed
and you loof^atyour watches, breakfast will still
be another hundred thousand years off.
Albert Engstrbm (1869 - 1940; Swedish artist and cartoonist)

31
Chapter
2
SYSTEMS OF NUMERATION
Page
2.0 Forms of Notation 3 2
2.1 Additive Notation 34
2.2 Multiplicative Notation 44
2.3 Positional Notation 46
2.4 Decimal Position System 48
2.5 Sexagesimal Numeration 56
2.6 Vigesimal Numeration 58
2.7 Duodecimal Numeration 61
2.8 Binary, Octal, Hexadecimal 62
2.9 Special Forms of Notation 66

32
Chapter 2 SYSTEMS OF NUMERATION
2.0 Forms of Notation
I cut every day a notch with my (qiife, and every seventh notch was
as long again as the rest, and every first day of the month as long
again as that long one; and thus I /^ept my calendar, or weekly,
monthly, and yearly reclining of time.
Daniel Defoe (1659-1731): The Life and Strange Surprising Adventures of
Robinson Crusoe of York, Mariner (1719)
Additive Notation
Subtractive Notation
Multiplicative Notation
Positional Notation
Place-Value Notation
We distinguish between three essentially different methods
of arranging numeral characters to form numbers: additive,
multiplicative, and positional notation.
Most ancient number systems used characters whose meaning
and value were always the same.
In additive notation, composite numbers are formed by
juxtaposing several single characters of each kind in a row,
generally in order of descending value, and repeating each
character as many times as required in order to form the
desired composite number.
To avoid number presentations of inordinate length, the
additive method of notation was sometimes supplemented by
subtractive notation, where a character of lower value
preceding one of higher value signifies that it is to be deducted from
the latter.
Of additive systems, we shall present Egyptian, Sumerian,
Greek, and Roman numeration.
In multiplicative notation, two kinds of number symbols are
employed, one kind being constant and its effective value
being increased by multiplication by a symbol of the other
kind, also of constant meaning and value.
The desired number is obtained by writing the multiplied
numerals in a row, and adding them up.
Of multiplicative numeration systems, we shall describe the
Chinese system, which appears in several forms, each with its
unique sphere of application.
Positional notation, or place-value notation, is the method used
exclusively by all modern number systems; it is a refinement
of the multiplicative system, having abolished the value
multiplier characters.

Section 2.0 Forms of Notation
33
A character in a positional numeration system retains its
meaning, but its actual value when part of a written number is
determined by its position, or place, in the sequence of
characters forming the composite number.
This will be illustrated by the Hindu-Arabic numeration
system, by the Babylonian and Mayan systems, and by the
computer-oriented binary system and its associated octal and
hexadecimal forms.

34
Chapter 2 SYSTEMS OF NUMERATION
2.1 Additive Notation
Egyptian Numeration
Egyptian hieroglyphics of some 5500 years ago are a
numerical grouping system whose symbols were used to signify
numbers which, when added together, could express any
desired number.
The number 1 was represented by a vertical line, or a picture of
a staff; 10 by a heel-bone sign; 100 by a coiled rope; 1000 by a
lotus blossom; 10 000 by a bent finger; 100 000 by a tadpole;
1000 000 by a kneeling genie with raised arms:
n Q
10
10
103
)
10
105
101
Latin, decim, "ten"
Greek, deka, "ten"
Latin, denarius, "containing ten"
Each symbol could be repeated up to nine times, signifying
addition. In a system of this kind the order of the symbols is of
no consequence, but the Egyptians usually wrote the symbols
in order of descending value, either from left to right or from
right to left.
1
1111111*«™"""™
nr\
A number system that uses ten as a base is called a decimal
(decadic, denary) system; thus, the notation used by the
ancient Egyptians is a strictly additive decimal notation.
The Egyptians generally used only so-called unit fractions
having the number 1 as numerator, which they wrote by
placing the symbol for an open mouth above the denominator:
nm
12)
1
QQOO 224-
llll
II *™
2160
Special symbols were used for certain fractions:

Section 2.1 Additive Notation
35
Sumerian Numeration
The Sumerians, who lived from about 3200 B.C. on the lower
reaches of the river Euphrates, early developed a high level of
culture and one of the oldest written languages known to us.
They had two systems of numeration: a sexagesimal system
for astronomical observations, and a decimal system for
everyday use. Both systems were additive: every symbol was
repeated the number of times required by the desired number.
The characters were made with rods of two diameters which
were pressed, at an angle or end-on, into tablets of clay which
were then sun-dried or kiln-baked.
10
60
10 x 60 602 10 x 602
Cuneiform Symbols
Latin, cuneus, "a wedge"
A subtraction symbol was sometimes used in order to reduce
the number of symbols required, and to save space:
O 7^ D
10-1
T*
4x600
oo
oo
4 x 10 = 2360
From about 2750 B.C., these archaic symbols were gradually
abandoned in favor of characters made with a sharp-edged tool
which left wedge-shaped impressions in the clay; these
cuneiform symbols were later traced with a fine stylus.
r
TT
TTT T Yf W ^ ?
8
<
10
«
20
«K<
30
<
40
«
50
$*"Y
60
10x60
T<
70
T«
80
J<«
90
The symbol denoting 602 = 3600 consisted of four wedges
arranged as the sides of a square "60 squared". For 10 x 602 =
36 000, a "corner" wedge denoting a factor of 10 is placed inside
the symbol; for 603 = 216 000, a large vertical wedge, denoting
a factor of 60, is placed inside.
602 = <^>
10x602
-<£
603
-<5>

Chapter 2 SYSTEMS OF NUMERATION
Akkadian Numeration
Around 2350 B.C. the Sumerians were invaded by the
neighboring Akkadians, who assimilated the major part of the
Sumerian culture, including their cuneiform characters
which they adapted to the needs of their own language.
Like most Semitic peoples, the Akkadians were used to
counting with hundreds and thousands, for which they found
no suitable characters in the Sumerian numeration system,
and had to devise new ones based on Sumerian cuneiform
characters. Thus,
Sumerian y "^ = 60 + 40 became Akkadian Y **- = 100;
for 1000, they wrote 4 y^_ = 10 x 100.
Greek Numeration
The Phoenicians had special characters for numerals which
the Greeks did not adopt when they copied the alphabet. They
used, instead, various forms of alphabetic notation for
numbers.
In an early form of numeration some letters were assigned a
numerical value, but this was found to be impractical for
mathematical operations and was replaced by two major
systems of numeration in ancient Greece.
Attic Numeration
Athens and the surrounding province of Attica used a
numeration system in which
I stood for unity,
T, an old form of n (pi), for 5 dlevTe), and
H (eta) for 100 ('£koctov),
while other numbers were denoted by the initial of the name of
the number, thus
A (delta) for 10 (Aeica),
X (chi) for 1000 (XiAaoi), and
M (mu) for 10 000 (Mupioi).
The symbols were combined in the following way:
i-A i-H rX i-M
50 500 5000 50000

Section 2.1 Additive Notation
37
The Attic system is sometimes called Herodianic, named
HERODIAN after Herodian, by profession a grammarian, who gave a
(2nd century) description of the system toward the end of the 2nd century,
long after the system had, in fact, gone out of use. Examples of
its use can be found in Attic inscriptions from 454 to
about 95 B.C.
Ionic Numeration
The Ionic system of numeration was a more suitable system
for mathematical operations. There is evidence that the
system was firmly established in the 8th century B.C.
The system's name refers to its origin in Ionia, the ancient
Greek colony in Asia Minor, established around 1000 B.C.,
today part of Turkey.
For the numeration system, three otherwise obsolete symbols -
digamma, koppa, and san - were revived and added to the
Classical Greek alphabet. The twenty-seven characters thus
obtained were divided into three groups, denoting units, tens,
and hundreds:
A, a
B, P
r.y
A, 8
E, e
C,C
z,C
H, Tl
©, e
alpha
beta
gamma
delta
epsilon
digamma
zeta
eta
theta
1
2
3
4
5
6
7
8
9
I, i
K, K
A, X
M, \i
N, v
3.*
O, o
IT, n
9,?
iota
kappa
lambda
mu
nu
xi
omicron
pi
koppa
10
20
30
40
50
60
70
80
90
P, P
£, o
T, %
T, A)
O, <p
x, x
^, ¥
Q, ©
^,¾
rho
sigma
tau
upsilon
phi
chi
psi
omega
san
100
200
300
400
500
600
700
800
900
It was common practice to use a horizontal bar above letters
denoting numbers, to distinguish them from the surrounding
text.
The numerals were written according to the additive
principle, starting from the left with the higher denominations,
PKA ONE yOZ QHT
and
pic8 cpve ij/o£ cony
124 555 777 883
A mark at the upper left of the unit letters denoted thousands:
'A 'B T ... '0
1000 2000 3000 9000
«

Chapter 2 SYSTEMS OF NUMERATION
Ten-thousands and higher numbers were denoted by the letter
M (mu) - for Mtipioi = 10 000 - with a letter or group of letters
signifying the number of ten-thousands wanted placed above
it:
a _B y 0
M M M ... M
10 000 20 000 30 000 90 000
M M M
990 000 9 990 000 99 990 000
These are merely examples of methods used by the ancient
Greeks to express large numbers. Geographic variations
occurred along with considerable ingenuity and individual
preferences among writers.
Although the ancient Greek way of expressing large numbers
by certain combinations suggests the possibility of a positional
system of numeration, both the Attic and the Ionic notations
are strictly additive decimal systems of numeration.
Fractions were written in various ways by the ancient Greeks,
largely according to personal preferences. The famous
mathematician and physicist-inventor Archimedes used an
ordinary cardinal number for the numerator, followed by an
accented number representing the denominator, e.g.,
Ancient Hebrew and Arabic Numeration
The use of alphabetic characters in the Hebrew system of
numeration evolved from the Ionic system. The oldest finds of
the Hebrew system date from no farther back than the 1st
century B.C., while Ionic numeration goes back to the 8th
century B.C.
After the Hebrew language had adopted the Ionic model of
numeration, the system was also adopted into the Arabic.

Section 2.1 Additive Notation
39
Roman Numeration
Roman numerals were modeled on the ancient Greek system,
using letter symbols for powers of ten and for intermediate
"fives". The primary symbols still in use today,
I
1
ten
/
1
V X L C D
5 10 50 100 500
l over, in part, from the Etruscans:
A A 7 C
5 10 50 100
M
1000
The symbols D and M were not part of the original system of
symbols. The number 1000 was initially written either CIO or,
in a simplified form, (I); the symbol denoting 500 was
obtained by splitting CIO down the middle and using either of
the halves.
Starting with CIO, every additional CO or ( ) bracketing a
number increased its value 10-fold:
CIO
(I)
1000
CCIOO
((1))
10 000
CCCIOOO
(((I)))
100 000
CCCCIOOOO
((((I))))
1000 000
Printers and stonemasons simplified these symbols further to
D = 500 and M = 1000.
A German printer's guide published in 1835 has these rather
lax rules for writing large numbers:
j£)anbburl)
. . . 1,000
in
BnclfbruckcrUntist.
M
CI3
JJCJ3 ...8,000
JIICI3 • • • 3)000
10. Qiiprr.
Carlsruljr »»> Caftrn,
h In D. E. A«ri'»<fci« 0«uM««fcUa|.
1836.
XCI3
CCJ33
3MC
liMI
CCCI333
CM
CCM
DCCCCM
CCCCI3333
. . 10,000
100,000
200.000
900,000
1,000,000
There were also other ways of increasing the value of a
number symbol. Bracketing a symbol by two vertical lines
increased its value 100-fold; a bar placed above a number
meant a 1000-fold increase of its value; a combination of both,
a complete "doorframe", above a number increased its value
100 000 times:
|x| x [x]
1000 10 000 1000 000

40 Chapter 2 SYSTEMS OF NUMERATION
/482 -J&ft
Only capital letters were used by the ancient Romans to
represent numbers; lowercase letters came into use for this
purpose in medieval times.
Additive Notation
To write other numbers, the Romans used the additive
principle and simply ran the requisite number of symbols of
each kind together in order of descending value, as in this
reproduction from 1888 (Paul Krueger, Berlin) of part of the
JUSTINIANUS table of contents of the law book of Emperor Justinian I, the
Petrus Sabatius Flavius Corpus Iuris Civilis, compiled in the middle of the 6th century:
LIBER PRIMUS XXXVII De officio praefecti Augustalis
XXXV111 De officio vicarii
XXXVllll De officio praetorum
XXXX De officio rectoris provinciae
XXXXI Ut nulii patriae suae administratio sine
speciak permissu principis permittatur
XXXXU De quadrimenstruis tam civilibus quam
muitaribus brevibus
XXXX111 De officio praefecti vigilum
XXXXUU De officio praefecti annonae
XXXXV De officio civilium iudicum
XXXXV1 De officio iudicum militarium
XXXXV11 Ne comitibus rei militaris vel tribunis
lavacra praestentur
XXXXV111 De officio diversorum iudicum
XXXXVUU Ut omnes tam civiles quam militares
iudices post administrationem deposi-
tam per quinquaginta dies in civita-
tibus vel certis locis permaneant
L De officio eius qui vicem alicuius iudi-
cis obtinet
LI De adsessoribus et domesticis et can-
cellariis iudicum
Subtractive Notation
To avoid fourfold repetition of a symbol, the subtractive
principle of writing was introduced. Placing a symbol of
a small value immediately in front of one of higher value
implies that it is to be deducted from the higher number so that
the combined symbol signifies the difference between the
higher and the lower numeral; thus
IV
4
instead of
IIII
IX
9
Villi
XL
40
XXXX
xc
90
LXXXX
CD
400
cccc
CM
900
DCCCC
The subtractive principle was subject to two rules:
o The symbols V, L, and D must not be used as the number to
be subtracted;
o Only one symbol I, X, C may be placed before a higher
number symbol.

Section 2.1 Additive Notation
41
SCHEDEL Hartmann
(1440-1514)
Thus XXC was not permitted for 80, but these rules were not
strictly obeyed, particularly for higher numbers, as we have
seen in Herr Hasper's handbook; and it is not uncommon to
see in older texts, for instance, IIX denoting 8.
The sub tractive principle was introduced in the 13 th century
but took a long time to gain acceptance, as witnessed by
an excerpt from Buck der Cronicken (Nuremberg, 1493)
by Hartmann Schedel - physician and hagiographer - which
also shows an alternative method of writing the calendar
year:
Jav bet Tct&.vpi.vfMvL 3*v Czffi?a-*W-&*>$*
Djkoiax)* 6cr funfr 6iTOO* O>oma* fm^im^ gc
•fchlcci>r geporn tins argte fan xcatb mtt gemaynct foig babft crfom ;n ban
mortmlTIaixqn^&ctgcpn^ foicnet 6ebfHs
d>q: bihfc vnb crcn xco\ ToitbiQ.l& waa olfo miltvnnb b>ct bic gcicttcn maii olfo
lic6 bat ctbicfdbtn $o ambttn vnb pfv&nbtn rvunbapcdxcb gecn fdrfcerc, vnb
ficvmbirenetytcnvnbtulrnctfchcnbtsfcrkd)ifd)cngcjdingatnb*»\attmwoi be
lonct.alfo ba* bit tncrf>ifcb fct^ifr 6u bey fccfypfymxbttx iacn pecboxgen gclcg? wj
wibcrmn& in 6aa iicd)t gcmad>r xoaibt.l£r fctyidtt aucfy gderr man in aiics X£u'
ropam a»§ $cfiid>cn 6u Weeper 6ic aug pcrfawmnn* 6 a citcn rnb buret) abnl/
gang 6cr Catttm vnb rngiawbigen groben voids v ergangen watnX>nb nacfy
KieoLm* bet fBnft
9fcar 0/ f/k 74^/¾ vim • vic • xlvi (6646).
Jear of Christ, jm • iiijc • xlvij (1447).
9\(icholas the Jifth, formerly Iqiown as Thomas of Sarzana, of low
station and lineage, born a physician's son, was unanimously
elected Pope in the month of March, in the year 1447 of Our Lord.
He was well worthy of Papal Tminenee and Qlory.
In his clemency, he held learned men so dear that he was
wondrously proud to e?(alt them to high offices and prebends, and
he rewarded them well in recognition of their rendering of the
Qreefi^ tongue into Latin, so that Qreefi^ te?(ts that had lain
incomprehensible for si?t hundred years were again brought into
light
He also despatched learned men into all "Europe to search for boofe
that had been lost through neglect on the part of forefathers and
through the abuse by the Infidel and the coarse Heathen.
The "Year of the World" tells us that, by Papal Decree, the
world was just 5199 years old at the time of the Star of
Bethlehem.
We note the practice of substituting j for the final i in groups of
more than one i, in order to minimize the risk of misreading:
the single j of the thousands in Jar Cristi is used for the same
reason: to distinguish it from the final i in Cristi.

Chapter 2 SYSTEMS OF NUMERATION
In old ledgers of the Scottish Exchequer from the early 1300s
one may find the sum of, say, £8697 14s. 8d.
m c xx
vnj vj nij xvj fij xmjs vnja
8ooo + 6oo + 4x2o + 15 + 2£ + 14s + 8d
which also reflects Celtic counting by twenties,
och mile si a ceud ceithir fichead seachd deug
eight thousand six hundred four twenties seven-ten
Fractions
Although the Roman numeral system was generally decimal,
the method of representing fractions was based on the number
twelve. Various systems of notation with groups of dots or
lines were developed; in medieval times S was added to denote
semis (one-half):
ll x s
JL _L_I JL_I ±_I JL iL_I
12 12"6 12 " 4 12~3 12 12 " 2
s- s- s— s:: sx i
12 12"3 12"4 12~6 12 12 " 1
Intractable Surrender
Roman numerals were used in bookkeeping well into the 18ih
century and beyond because of the ease of adding and
subtracting written numbers by just keeping in mind that
five I make V
two V make X
ddition
M CCLXXX II
DC VI
MDCCCLXXXVIII
five X make L five C make D
two Lm
1282
+ 606
1888
ake C two D make M
Subtraction
MCCLXXXII 1282
DC VI - 606
DC LXXVI 676

Section 2.1 Additive Notation
43
Roman characters were, however, cumbersome to write and
clearly unsuitable for forms of calculation other than addition
and subtraction. Perforce, they had to give way to the Hindu-
Arabic numerals that we use today.
This was a long drawn-out process, however.
The city of Florence, in 1299, issued an ordinance concerning
the use of the "new numerals", pointing out with what ease they
could be falsified in the entries in the ledgers of the city and in
tradesmen's books: a 0 might be made into a 6 or a 9, and a 1
into a 4, 7, or 9.
In 1348, a statute of the University of Padua ruled that the price
lists of the University's stationarii (booksellers) must be
clearly written "non per cifras sed per literas claras", that is to
say, with Christian characters, by which were understood
Roman numerals.
The use of Roman numerals and the abacus is described in
p. 168 Chapter 4.
Roman Revival
Roman numerals enjoyed a kind of revival in the 1920s. The
Roman numeral X for 10 resembled the ends of a sawhorse,
and a ten-dollar bill was called a sawbuck, or saw for short,
with a one-dollar bill simply called a buck. A twenty-dollar
bill then naturally became a double saw, and a five-dollar bill
a half-saw.
A hundred-dollar bill was called a C-bill, a C-note, or just a C;
with a departure from Roman practice, a thousand-dollar bill
was known as a grand, or a G.
The idioms buck, C-bill, C-note, G, and grand are still
perfectly good linguistic currency throughout the United States.

Chapter 2 SYSTEMS OF NUMERATION
Multiplicative Notation
A multiplicative numeration system has two kinds of
numeral symbols. By a multiplicative process, the characters
of one kind of symbol increase the value of the other kind. The
number is then expressed by the multiplied numerals, being
added together.
For instance, using 1 through 9 to represent these values, and
the letters A, B, C, and D to represent 10, 100, 1000, and 10 000,
respectively, we can write
4D 5C 6B 1A 2 = 45 612
Chinese Numeration
The oldest discoveries of a Chinese written numerical system
are from the Yin dynasty (c. 1523 - c. 1027 B.C.), but the system
may be considerably older.
Japan, Korea, and several other countries in the Orient have
adopted Chinese numeration.
We can distinguish at least three different styles of Chinese
numerical symbols: traditional national numerals, official
numerals, and mercantile numerals.
Traditional National Numerals
The traditional Chinese system of numeration is a decimal
system employing thirteen basic characters, expressing the
numbers 1 through 9,10,100,1000, and 10 000:
123456789
+ S =t Hor7f
10 100 1000 10000
These symbols, the traditional national numerals, are the
symbols most commonly used today. They originated under
the Han dynasty (206 B.C. - A.D. 220) and have kept their
structure for more than 1700 years.
Traditionally, Chinese numerical signs were arranged
vertically from top to bottom, but horizontal alignment from
left to right is common practice today.
8 x 10000 + 9 x 1000 + 5 x 100 + 6 x 10 + 7 = 89567

Section 2.2 Multiplicative Notation
45
Official Numerals
The official numerals are beautifully ornamented forms
used on banknotes, bonds, deeds, contracts, or other valuable
documents as a safeguard against forgery:
■£• m ^ n & m % w> \\
123456789
» « ff w
10 100 1000 10 000
Mercantile Numerals
Mercantile numerals, traditionally used by shopkeepers, are
falling into disuse, but can still be found both in Mainland
China and in Taiwan.
o i u m y frj.±± #
012345678 9
+ 3ox2? ^ 7*
10 100 1000 10000
To denote zero (Chinese, ling), the zero sign of the mercantile
numeral system is frequently used today with the traditional
national system.
Besides the styles of Chinese numerals described above, there
are cursive forms used for handwriting, as well as purely
calligraphic styles.
Chinese rod numerals will be discussed in their appropriate
place, under Section 2.4, Decimal Position System.

Chapter 2 SYSTEMS OF NUMERATION
Positional Notation
blessed are the, placemakgrs: for they shaft be catted the children
of Qod.
Bible (1562 misprint); Matt. 5: 9
Names of Systems
In a positional numeration system (or place-value system), the
numerical value of each symbol, or digit, depends on its
position in the row of digits.
Every integer larger than 1 can serve as a base in a positional
numeration system, and the system requires exactly as many
digit symbols as indicated by the numerical value of the base.
Positional numeration systems are named after the
numerical value of the base.
Base
2
3
4
5
6
7
8
9
10
11
12
16
20
60
Name of System
binary (dyadic)
ternary
quaternary
quinary
senary
septenary
octal (octenary)
nonary
decimal (decadic, denary)
undenary
duodecimal (duodenary)
hexadecimal (hexadecadic)
vigesimal (vicenary)
sexagesimal
A Peep into Prehistory
Schoo[children in the author's native Sweden used to read of a
caveman and hunter named cLira-%aipa. % the best of the author's
recollection,
... clira-%aipa was the immensely proud owner of no less than
twenty-seven stone a?(es. Of course, he didn't actually (qiow in so
many words that there were eT^actly twenty-seven of them, because

Section 2.3 Positional Notation
47
he could not count beyond three, but that did not trouble him at
alt.
He was a reasonably friendly old body with a neat and orderly
mind, who lilted to feep tabs on his goods and chattels, quic/^ and
dead, Assuredly, he could not write, because writing had not yet
been invented, so he did the ne?(t best thing.
He arranged his axes in little groups of three, three to a row, and in
three rows, life this:
TTT til TTT
ft? ITT Iff
TTT lit TtT
so he could see at a glance at an audit if they were all present and
correct, or if somebody had perhaps nicfed one.
9{pt being a man of great learning, as we have said, he was
blissfully unaware of the fact that he must have been the inventor
of the ternary system of counting. If he had Iqiown Hindu-Arabic
numerals as we Iqiow them, he would have totaled his hoard as
10 00 axes
As we realize, twenty-seven was the ma?(imum number of stone
axes or anything else that anyone could possess — because what
would you do if you had one more? Jou couldn't count it in the
established manner, it just wouldn't fit in anywhere - in fact, it
would constitute the closest approximation to infinity, and you
would simply have to throw it away, wouldn't you?

48 Chapter 2 SYSTEMS OF NUMERATION
2.4 Decimal Position System
Hindu-Arabic Numeration System
We use the base ten, or decimal, system daily, so it is the
natural place to start if we want to understand positional
systems generally.
Each of the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 represents a number on
its own. The decimal positional system gives an
unambiguous meaning to strings of digits as a sum of powers of ten, the
rightmost digit being the number of ones (where we let 1 = 10°).
The next digit to the left is the number of tens, so one ten and
zero ones is written 10.
One of the wonders of this system is that while there are two
ways to read 2300 (two thousand three hundred or twenty-three
hundred), both name the same number. The canonical way to
interpret 4321 is
4 thousands + 3 hundreds + 2 tens + 1 one
= 4xl03 + 3xl02 + 2xl01 + lxl0°,
but we get the same result from
43xl02 + 2xl01 + lxl0° or 432x 101 + lx 10°.
The convention forced upon us by the desire to make the
algebra of exponents consistent is that a0 = 1 whenever a^0.
The decimal system or any other positional numeration
system may be extended beyond the integers to represent
fractions. A point of reference is then selected to establish the
positional values of the several digits, and characterized, for
example, by a decimal point:
310.103 = 3xl02 +1x101 + 0x10°+1x10-1 + 0x10-2+3x10-3
Origin and Development
The oldest preserved samples of Hindu-Arabic, or Arabic,
digits are inscriptions on a stone column found in India and
believed to be from around the year 250 B.C. However, this and
other early specimens display no zero and give no indication
of the present-day use of Arabic digits. Scholars are uncertain
about the earliest date for the use of these early digits and zero
in a positional numeration system and whether the Hindu
positional system developed entirely within India or under
Phoenician and Persian influence.
al-KHOWARIZMI The oldest known writing using a fully developed numer-
Muhammad ibn Musa ation system appears in a book from A.D. 825 by the Persian
(c. 780 -c 850) mathematician al-Khowarizmi. The original is lost, but a

Section 2.4 Decimal Position System
49
Latin translation from around 1120 exists, Liber algorismi de
numero indorum ("The Book of al-Khowarizmi on Hindu
Number") by the Englishman Robert of Chester. In this
translation, the numerals were erroneously assumed to be of
Arabic origin, a mistake carried through into present-day
terminology. The European writing of the digits was
influenced by Arabic manuscripts, but their European design also
followed paths independent of Arabic models.
The importance of the 12th-century Latin translation of
al-Khowarizmi's book on numerals is witnessed by the term
algorithm (or algorism), a Latin corruption of al-Khowarizmi,
which came to mean the art of computing with Hindu-Arabic
numerals. Today algorithm refers to any systematic
mathematical procedure that leads to a solution usually in a finite
number of steps.
Based on the Hindu model, the Arabs developed two versions of
numerals, East Arabic and West Arabic, of which the East
Arabic is now by far the dominant. The numerals below
represent modern East Arabic numerals:
0123456789
Although the Arabic language is written from right to left,
Arabic numerals are arranged as in English, e.g.,
50 65 1748 1996
In Arab countries where the East Arabic numerals are
prevalent, visitors are often surprised that the European
version of "Arabic numerals" bears so little resemblance to the
indigenous numerals. This is because European Arabic
numerals were modeled primarily on the now extinct West
Arabic numerals, e.g.,
which varied considerably, however, geographically and with
time.

Chapter 2 SYSTEMS OF NUMERATION
Their name, Gobar, is Arabic for sand board or dust, referring
to the practice of performing calculations directly on the
ground or on a board covered with sand, which could be wiped
clean when a calculation was completed. The name is now
attached to the early Hindu-Arabic numerals introduced into
Europe. The oldest surviving sample of Gobar numerals in
Europe is a Spanish manuscript from 976:
i i i ? v l i a j
123456789
Leonardo Fibonacci (Leonardo of Pisa) was an Italian
merchant traveling in the Orient, where in addition to conducting
business he also learned Oriental mathematics. Fibonacci
wrote the Liber abaci (1202) - in all likelihood inspired by
al-Khowarizmi's writings - and Practica geometriae (1220).
Liber abaci, dealing with arithmetic and algebra, played an
important role in spreading the knowledge and use of the
decimal positional system in Europe.
Significant Digits
The digits that define a numerical value are known as
significant digits.
The significant digits of a given number
begin with a first non-zero integer digit or - if the
number is less than unity - with the first, zero or
non-zero, decimal digit;
and end with the final, zero or non-zero, decimal digit; the
final zero or zeros of an integer (whole) number
may or may not be significant.
Thus,
3200 has at least 2 3.20 has 3
320 has at least 2 0.320 has 3
32 has at least 2 0.0320 has 4
32.0 has 3 0.032 has 3
significant digits.
To make clear the number of significant digits of an integer
p. 162 ending in one or more zeros, it can be written in scientific
notation. For instance, to clarify that 2304 000 has only four
significant digits, we write 2.304 x 106; to clarify that all digits
are significant, we write 2.304 000 x 106.
A number with four significant digits is said to have a four-
place accuracy.
50
Gobar
FIBONACCI Leonardo
(c. 1170-c. 1250)

Section 2.4 Decimal Position System
51
Chinese Rod Numerals
The rod-numeral system represents a positional decimal
system created in China at least 2000 years ago, which uses
bamboo rods to represent numbers. The system was adopted by
Japan and Korea and used for arithmetic and algebraic
calculations until about 150 years ago.
The numerals 1...9 can be reproduced by two different series
of rod configurations:
First series
II III
T TI HH
8
Second series
123456789
Writing the number 21 as ill or ^zr would risk confusion with
3; written as /f I it would be read as 201; with two series of
symbols available, we can write
U- or =1
which are unambiguous.
For many centuries the system lacked a sign for zero. Simply
leaving a space between symbols did not ensure unequivocal
discrimination between numerals such as 36, 306, 3006, 3060,
and 3600.
To overcome this difficulty, a traditional Chinese number
symbol was sometimes used as a separator; at other times, a
reckoning board was used, an empty square clearing up the
matter, e.g.:
r
0
0
w
0
0
T
Through Indian influence in the 13th century, a symbol for
zero - a circle - was incorporated into the rod-numeral
system, making all numbers, and particularly fractions,
easier to express (a leading zero can be assumed to designate a
decimal marker):
o
0.15
oo o J
0.0079

52
Chapter 2 SYSTEMS OF NUMERATION
Metric System
International System of Units, SI
The Decimal System of Measurement
To get the full benefit of a positional numeration system, the
system of units of measurement must agree with the
numeration system.
The term decimal system is not limited to systems of
numeration that employ base ten, but is also used to
characterize systems of units and measurement whose multiples and
submultiples of units are powers of ten. The metric system
is based on multiples of 10 and the International System of
Units (SI), which is convenient for science and technology, is
organized around multiples of 1000 = 103.
With the major exception of the United States, SI is today used
almost exclusively in the industrialized world; it is also the
standard of most developing countries. Aside from the basic
sciences and some major industries, the U.S. has been
resistant to adopting the metric system and its successor, the SI.
For all the technical and scientific advancement associated
with the U.S., it is bizarre that it so staunchly abides
by a conglomeration of basically incoherent measurement
systems.
This is even more surprising since, in 1790, Thomas
Jefferson, then secretary of state, suggested that a decimal
system of measurement would be a way of achieving national
uniformity of weights and measures. In 1875, the U.S. was one
of the original signatories of the Metric Convention, and since
1893 the definitions of the U.S. foot and pound rely on
the metric system; since 1959, a U.S. foot is defined as 30.48
centimeters.
"An unprecedented blizzard has
buried the Eastern seaboard under
one hundred and fifty-eight inches of
snow. That's, let's see ... twelve into
fifteen goes once, two from five is
three, bring down the eight, twelve
into thirty-eight goes three times with
two left over ..."
-^¾^¾
Drawing by H. Martin; © 1987
The New Yorker Magazine, Inc.
Will he manage before the next thaw?

Section 2.4 Decimal Position System
53
Scientific Notation
The handling of very large and very small numbers is
always fraught with difficulties.
o The radius of the Earth is 6.37 million meters, and the
height above Earth of a geostationary communications
satellite is some 36 million meters; these distances are not
too shocking to the reader.
o An astronomical unit - the mean distance from the Earth to
the Sun - is about 150 thousand million meters, or more
exactly,
149 569 800 000 meters.
o The diameter of the Milky Way, the galaxy in which our
Earth is an insignificant speck of dust, is on the order of 100
trillion (U.S.: 100 quintillion) meters - a 1 followed by 20
zeros.
o At the other end of the scale, the diameter of a poliomyelitis
virus is some 28 thousandths of a millionth part of a meter;
that of an atomic nucleus may be about 75 thousand-
billionths (U.S.: 75 quintillionths) of a meter:
0.000 000 028 or 0.0728
0.000 000 000 000 075 or 0.0^75
Whatever way we choose to write these numbers, they are
unwieldy and may easily lead to miscalculations.
To avoid these difficulties, we observe that every positive real
number a may be written in scientific notation,
a = p • 10"
where n is an integer, positive or negative; and 1 <p < 10, that
is,p is larger than or equal to 1, but less than 10.
To simplify calculation, specific prefixes to denote certain
multiples and submultiples of units of physics have been
introduced by the
General Conference on Weights and Measures
(CGPM, Conference Generate des Poids et Mesures), which
was founded by the Metric Convention of 1875. The majority of
these prefixes date from the 1960 conference; in October 1991
yotta, zetta, zepto, and yocto were added.
The prefixes form part of the International System of Units,
or SI (Systeme International d'Unites), which is, in fact, a
modernized metric system.
By SI recommendations, prefix symbols should be in Roman
(upright) type and attached to the SI unit symbol, e.g.,
millimeter (mm), microampere (|llA), decibel (dB).

54
Chapter 2 SYSTEMS OF NUMERATION
The multiple should be chosen so that the numerical value
preceding it will be between 0.1 and 1000. In tables, and in
contexts where direct comparison between values is made, it
may be advantageous not to change prefixes.
SI Multipl

Multiplier
10 24
1021
1018
1015
1012
109
106
103
102
101
io-1
10-2
IO-3
10-6
io-9
io-12
10-:i5
10-B
IO"21
io-24
*
*
*
*
es
SI Prefix
Name
yotta
zetta
exa
peta
tera
giga
mega
kilo
hecto
deca
deci
centi
milli
micro
nano
pico
femto
atto
zepto
yocto
Symbol
Y
Z
E
P
T
G
M
k
* h
* da
* d
* c
m
H (mu)
n
P
f
a
z
y
Etymology
Greek:
Greek:
Greek:
Greek:
Greek:
Greek:
Greek:
Greek
Greek:
Greek:
Latin:
Latin:
Latin:
Greek:
Greek:
Italian:
Danish:
Danish:
Greek:
Greek:
oktakis, eight times
heptakis, seven times
hexakis, six times
pentakis, five times
teras, monster
gigas, giant
megas, big
chilioi, thousand
hekaton, hundred
deka, ten
decim, ten
centum, hundred
mille, thousand
m tkros, small
nanos, dwarf
piccolo, small
femten, fifteen
atten, eighteen
heptakis, seven times
oktakis, eight times
(8x3
(7x3
(6x3
(5x3
(-21 = -
(-24 = -
= 24)
= 21)
= 18)
= 15)
7x3)
8x3)
* Not advocated in scientific and technical writing;
acceptable for domestic and lay use.
ANGSTROM, Anders Jonas
(1814-1874; Swedish physicist)
With the introduction of SI prefixes some older measurement
units have become obsolete, notably the angstrom (A) unit of
length which is equivalent to 0.1 nanometer or 100 picometers.
Bits and Bytes
Alphanumeric or alphameric characters consist of alphabetic
and numerical symbols, punctuation marks, and other
symbols used in computer work.
In computer terminology bit stands for binary digit. The bit is
the smallest storage unit in a computer.

Section 2.4 Decimal Position System
55
A series of consecutive bits representing one alphanumeric
character in the storage unit ("memory") of the computer
forms a byte (origin possibly from alteration of a bite, a small
piece). The number of bits that form a byte consists of eight bits
(eight on-off combinations, that is to say, a combination of
eight binary numbers), and each byte can thus represent one of
28 (= 256) combinations.
A kilobyte is 210 (= 1024) bytes and is denoted K. Kilo normally
represents a 1000 multiple of a unit; consequently, the term
kilobyte is not in line with SI or any other system of units.
A megabyte (MB) is 220 (= 1048 576) bytes, a gigabyte (GB)
230 bytes, etc.
Drawing by Levin; © 1989
The New Yorker Magazine, Inc.

Chapter 2 SYSTEMS OF NUMERATION
Sexagesimal Numeration
The Babylonian, or Mesopotamian, sexagesimal system of
numeration, which succeeded the Sumerian-Akkadian
system, is the oldest known example of numeration using
place value. We do not know exactly when this idea was
first conceived, but archaeological finds suggest it originated
around the beginning of the 2nd millennium B.C., with the
creation of the Babylonian Empire.
Several Orders of Unity
Our decimal positional system requires ten individual signs
to represent the numeral digits 0 through 9; a sexagesimal
positional system, using 60 as a base, would need 59 different
characters for the numerals within each span of sixty.
The Babylonian scientific numeration system is not a strictly
positional system. It uses the additive principle of the
Sumerians for all numbers below 60, and makes do with just
two characters: a "corner" sign <( signifying "ten" (10),
and a "wedge" sign V for all forms of unity: 1, 60, 602, 603, etc.
Thus, a number below 60 is written, e.g.,
47= £< $? (40+7)
and a number greater than 60,
78= Y^S? (60 + 10 + 8)
vv
The Babylonian number
or, transcribed into Hindu-Arabic characters,
1; 54; 42; 40,
means
1x60s = = 216 000
54x602 = 54x3600 = 194400
42X601 = = 2520
40x60° = 40
412 960
The Appearance of Zero
The Babylonians originally had no zero so they could not
mark the place of a missing power of sixty in a number, which
caused the same uncertainty as we would experience today

Section 2.5 Sexagesimal Numeration
57
without a zero to help distinguish between numbers like 61, 601,
and 6001, for instance. The only means open to a Babylonian
scribe was to leave an empty space where a zero would be.
An empty space has the disadvantage, however, of not being
easily recognized as a single, double, or perhaps multiple
empty space; even less is it identifiable at the beginning or
end of a number. Babylonian mathematicians and scientists
had to either invent some distinct, unmistakable place marker
or invent zero. They used a double "corner" sign or a slanting
double "wedge" as a place marker:
<* : * :^ -.4 ;\
which could be used equally well in an initial, intermediate,
or terminal position.
On tablets from the late 4th century B.C. - and believed to be
copies of documents written some two centuries earlier - we
encounter another character,
a corner sign with an elongated lower shank or tail, probably a
scribe's shorthand version of a double corner sign, with one
corner left out.
TT <«
2; 0; 27
which in our way of thinking is
2x602 + 0x601 + 27 = 7227.
This "zero" sign, from about 500 B.C., in many respects
functions as our zero, but the Babylonians probably had no concept
of zero as a number and instead used this symbol to mark a
void.
Sexagesimal Fractions
The use of a double corner or a double wedge in an initial
position provided the Babylonian astronomer and
mathematician with a simple means of writing sexagesimal
fractions (that is, fractions whose denominators are powers of
sixty), important to Babylonian science.
01 a si ,s/ 0 0 30
^T -60°+60l ^^ ^ " 60° + "60l+60*
0; 1 0; 0; 30
Such sexagesimal fractions survive to our day: an hour
has 60 minutes, each of 60 seconds; a full circle has
6 x 60 = 360 degrees, each divisible in 60 minutes of arc, each of
which is 60 seconds of arc.

Chapter 2 SYSTEMS OF NUMERATION
Vigesimal Numeration
The vigesimal numeration system was the usual mode of
counting in many ancient cultures because humans have ten
fingers and ten toes on which to count. As we know from the
Celtic languages and their remnants in French and Danish,
vigesimal numeration has a long endurance, as attested by the
lingering use of the word score.
Decimal-system numbers 2310 and 128 would be, in vigesimal
notation,
2310 = 5 x 202 + 15 x 201 + 10 x 20° = five; fifteen; ten
128 = 6 x 201 + 8 x 20° = six; eight
Mayan Numeration
The Maya Indians lived in Central America - in the Yucatan
Peninsula and neighboring Guatemala and Honduras - from
perhaps 200 B.C. until A.D. 1540 when they disappeared from the
\ pages of history.
, During the formative period of their civilization - the first
' three centuries of our time - the early Maya seem to have
I devised a purely additive vigesimal system of counting for the
everyday affairs of the general people. Indications of this are
| as many as they are vague, but as such trivial pursuits were
seldom committed to writing, there are no actual, tangible
'I proofs available of this, "the little man's arithmetic".
*#| The classical period of Mayan civilization - from about
A.D. 290 to 925 - is characterized by the attainment of a
|| high level of culture, with truly remarkable achievements in
arithmetic, astronomy, and art. Symbols relating to Mayan
si numeration, calendar, and deities have been understood
II for over a hundred years, but not until in the 1980s could
epigraphers and linguists put together the vast body of scien-
•|| tific work for a comprehensive deciphering of Mayan writing.
•|| For scientific purposes, the Maya had a written positional,
near-vigesimal system of numeration possessing a definite
!|| zero. Besides highly ornate hieroglyphic numeral characters,
depicting their gods, the Maya had a simpler system of three
ill basic symbols:
o dots of value 1
o bars with value 5
o a stylized seashell, generally painted red, for zero
«||| Numbers from 1 through 19 were made up of these bars and dots
in the most "economical" manner, with the bars either
horizontal or vertical: if the bars were horizontal, the dots
UK were placed above them; if they were vertical, the dots were
placed on their left.
Ill

Section 2.6 Vigesimal Numeration
59
The system was not strictly vigesimal, the second span having
a base 18 instead of 20; a Mayan number is written in column
form and reads from top to bottom:
• Ix20xl8x202 = 144000
0x20x18x20 = 0
13x20x18x20° = 4680
7x20 = 140
15x20° = 15
148 835
The seashell zero could be used in intermediate and terminal
positions of a number, but not in an initial position. The only
use for a zero in our initial position would be for the analogue
of decimal fraction notation, but the Maya knew only integer
numbers and had no concept of fractions.
The Mayan Calendar
The Maya had two kinds of calendar: a religious calendar of
260 days consisting of 20 cycles of 13 days each; and a secular
calendar of 365 days, in fact, a solar calendar of 18 uinals
("months'') of 20 days each, plus an extra 5 days added at the
end of the year.
The actual length of a solar year is the average time elapsing
between successive passages of the Earth through the same
point in its orbit around the Sun. Mayan astronomers were
fully aware of the error in the length of the solar year, inherent
in the official 365 days.
According to our latest measurements,
- the average solar year is 365. 242 198 days
- the Mayas calculated it to be 365. 242 000 days
- as against the Gregorian calendar's 365. 242 500 days
which makes the error in the Mayan measurement just under
two-thirds of that of the Gregorian calendar.
Mayan astronomers also observed the movements of the Sun
and the Moon, of Venus, and possibly also of Mars, Jupiter,
and Mercury; and they predicted solar eclipses with an
astonishing precision.
To appreciate fully the excellence of these feats of Mayan
astronomers, we must remember that they did not know how to
make glass and thus had no lenses or telescopes or other
optical equipment which we regard as indispensable in
astronomical work.

Chapter 2 SYSTEMS OF NUMERATION
Nor did they know the sandglass or the klepshydra (water
clock), or any other means of measuring times shorter than
the day, which they looked upon as the shortest unit of time
- hours, minutes, and seconds being unknown quantities.
Yet they did measure the length of the true solar day - that is,
the average time elapsing between two successive passages of
the Sun through the observer's meridian, just by observing the
shadow of a gnomon - the "hand" of a sundial.
MAYAN CIVILIZATION
AZTEC EMPIRE
HONDURAS
EL SALVADOR
NICARAGUA
Aztec Numeration
From A.D. 925 the Maya went downhill, perhaps because of the
clearing of the forests by burning and of overcultivating the
soil, or perhaps because of social unrest and wars, and their
culture declined. What remained was taken over by some of
their neighbors, among them the Aztecs, known for their ritual
human sacrifices. In the early 1500s, the Aztec Empire was
conquered and many of its people massacred by the Spanish
conquistadores.
We know their system of numeration through contemporary
Spanish translations of Aztec books, most well known of which
is the Codex Mendoza, named after Don Antonio de Mendoza,
the first viceroy of New Spain.
The Aztec system of numeration was strictly vigesimal, with
special characters for 1, a dot; for 20, a drawing of a hatchet;
for 202 = 400, a bird's feather; for 203 = 8000, something looking
like a string purse, etc.
• •
P P. EJP
•••3 20 21 40 400 421 8000

Section 2.7 Duodicimal Numeration
61
2.7 Duodecimal Numeration
Latin, duodecim, "twelve" The number twelve has set its mark on many aspects of our
Old French, dozaine, "twelve" environment: there are twelve months in a year, and a clock
dial shows twelve hours; twelve items make a dozen, and
twelve dozen make a gross; a foot is twelve inches.
When the Babylonians were faced with the need for a smaller
counting unit in their sexagesimal system, they settled for ten
instead of twelve as a subsidiary base. They continued to use
sexagesimal fractions, however, so the advantage afforded by
the fact that twelve is divisible by 2, 3, 4, and 6 did not carry
weight with them.
Dividing the day into twice twelve hours is not, as long
thought, of Babylonian origin; Otto Neugebauer has shown
that it was an Egyptian invention.
There are examples, in the late 19th century, of countries that
redivided their old-time foot of twelve inches into ten "new"
inches, before finally "going metric".
A really forceful - and final - argument against duodecimal
numeration is the fact that we have a well-functioning
decimal system.
An interesting combination of vigesimal and duodecimal
counting was offered by the old British monetary system, a
pound sterling being divided into twenty shillings of twelve
pence each - until it was scrapped in 1971.
- It is, of course, well known among poultry farmers that
Scandinavian hens are vigesimal and lay scores of
eggs, whereas English-speaking hens are duodecimal
and lay theirs by the dozen.

Chapter 2 SYSTEMS OF NUMERATION
Binary, Octal, Hexadecimal
Binary Numeration
The binary positional numeration system uses 2 as base, and
thus requires only two distinct symbols, say 0 and 1. In binary
numeration, ten will be written 1010, because
10= Ix23+0x22 + Ix21+0x2°
where 2 ° equals 1, by definition.
The use of a binary system would be cumbersome for everyday
manual calculation because of the inordinate length of most
numbers, but it is ideal for electronic computers whose
mechanical and electronic relays, such as transistors, know
only two states of operation - on/off, closed/open, yes/no,
true/false.
With 1 and 0 as operational characters, 1 will stand for
on (closed circuit, true) and 0 for off (open, false).
Binary numbers, known to Chinese mathematicians since the
5th century or earlier, were investigated and set into a serious
numerical system by the eminent German mathematician
Wilhelm Gottfried Wilhelm von Leibniz, to whom is attributed this very
unmathematic thesis of abstract theology:
God, represented by the numeral 1, created the
Universe out of nothing, represented by 0.
Binary Counting Rules
Addition and subtraction of binary numbers are governed by
the following rules:
0 + 0=0 0+1=1 1 + 0=1 1 + 1=10 (0, carry 1)
0-0 = 0 1-0=1 1-1 = 0 10-1 = 1 (borrowl)
The binary multiplication table looks like this:
0x0 = 0 0x1 = 0 1x0 = 0 1x1 = 1
and the division table is even simpler:
0/1 = 0 1/1 = 1; 1/0 and 0/0 are undefined
• Convert decimal number 23 to binary notation.
The relevant powers of 2 are
24 23 22 21 2°
16 8 4 2 1

Section 2.8 Binary, Octal, Hexadecimal
63
23dec = lx24 + 0x23 + lx22 + 1x2* + 1x2° = 10111
bin
Convert binary 110111 to decimal form.
We have
lx25+ lx24+ 0x23+ lx22+ 1x2* +1x2° =
32 + 16+0+4+2 +1=55;
hence, binary 110111 = decimal 55
Calculate and express the sum, difference, product, and
quotient of binary numbers 1011 and 101 in binary and
decimal form.
1011bin = 8 + 0 + 2 + 1 = lld)
lec
101bin = 4 + 0 + 1 = 5^
Sum
1011
+ 101
ioooobin
Product
1011
x 101
1011
0000
+1011
llOlllbin
11
+ 5
16 dec
11
x 5
5 5 dec
Difference
1011 11
- 101 - 5
110 bin 6 dec
Quotient
10.00110... bin
10111011.00000
-101
01000
- 101
110
-101
^•-^dec
5 111.0
-10
1 0
-10
0
0010
"Ou
"<»Mt)t,
//>/llO'jO|j
O,ttonon
Vlla'Oinn 0n'Uu%
lanoiai '*„,
'Mien i0»i»i»
Binary Man. Drawing by Dedini; © 1987
The New Yorker Magazine, Inc.

Chapter 2 SYSTEMS OF NUMERATION
Octal and Hexadecimal Numeration
In computer practice, octal and hexadecimal numbers are
often used to represent large binary numbers which may be
difficult to handle because of their length. The usefulness of
these forms of notation arises from the ease of conversion
between the number systems, as both eight (23) and sixteen (24)
are powers of 2, the base of the binary system.
The octal numeration system requires only eight digits, which
can all be supplied by Arabic digits (0, 1, 2 ... 7).
The hexadecimal numeration system needs sixteen symbols.
As the decimal system provides only ten basic number
symbols, the letters A, B, C, D, E, and F are used as additional
symbols to make up sixteen symbols.
Decimal
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
23
24
25
31
32
Binary
0
1
10
11
100
101
110
111
1000
1001
1010
1011
1100
1101
1110
1111
10000
10001
10111
11000
11001
11111
100000
Octal
0
1
2
3
4
5
6
7
10
11
12
13
14
15
16
17
20
21
27
30
31
37
40
Hexadecimal
0
1
2
3
4
5
6
7
8
9
A
B
C
D
E
F
10
11
17
18
19
IF
20
Note that octal and hexadecimal numbers of, say, 13 and 20 do
not read "thirteen" and "twenty", but "one-three" and "two-oh",
respectively.

Section 2.8 Binary, Octal, Hexadecimal 65
System Conversion
To convert a binary number to octal notation, divide the
number into groups of three digits (triplets), beginning with
the unity digit and working left - or both ways if the number
contains a decimal part. Incomplete groups are filled in with
zeros, and the several triplets are converted to octal notation.
To convert a binary number to hexadecimal notation, divide it
into groups of four digits (quadruplets) in the same manner,
and convert these groups, one by one, to hexadecimal notation.
• Convert binary 10111110 to octal, decimal, and hexadecimal
notation.
Binary row 010 111 110
2 7 6 = 276oct
2x82 + 7X81 + 6x8°
128 + 56 + 6 = 190dec
Binary row 1011 1110
BE = BEhexad
Hence,
10 111 110bin = 276oct = BEhexad = 190dec
• Explain how (a): 7 + 7 = 13; (b): 7 + 8 = 12; (c): 7 + 9 = 11.
a: 7 + 7 = 1 x II1 + 3 x 11° = 13 ; 11 is the base
b: 7 + 8 = Ixl31 + 2xl3° = 12; 13isthebase
c: 7 + 9 = 1 x 151 + 1 x 15° = 11; 15 is the base
Explain how 2 x 33 = 2000 .
2x3x3x3 = 2x33 + 0x32 + 0x31+ 0x3° = 2000; 3isthebase

66
Chapter 2 SYSTEMS OF NUMERATION
2.9 Special Forms of Notation
The Morse Code
MORSE Samuel F. B.
(1791 -1872)
This code uses a system of short and long signals - dots and
dashes - to send messages by sound, as flashes of light, or on
the telegraph. Combinations of dots and dashes make up
letters, punctuation marks, and numerals.
The original system was developed in 1838 by the American
artist and inventor Samuel Morse; a modification - the
International Morse Code - was devised in 1851.
1
2
3
4
5
period
6
7
8
9
0
comma
■ ■ ■ = 2017
■ ■ = 0.65
Braille
BRAILLE Louis
(1809-1852)
Braille is the written language of blind people, named after its
inventor, the French educator Louis Braille, himself blind.
Its characters are raised dots - read by touch - arranged in
simple combinations in a two-by-three pattern.
The first ten letters, A through J, will double as numeral digits
1...0 when preceded by a special numeral identifier sign.
• 9
• 9
9 • 9 • 9 9 9 9
9 * 9
O 9 • • ** ** **
Numeral Identifier A or 1 B or 2 C or 3
9 •
• 9
D or 4 E or 5
O O 9 9
9 • * 9
9 •
9 9
• •
Fcr6
• •
• 9
9 *
G or7 H cr8 I or9
• 9
9 9
JorO

Section 2.9 Special Forms of Notation
67
Manual Sign Language
d'EPEE Charles Michel
(1712-1789)
GALLAUDET Thomas Hopkins
(1787-1851)
Most national sign langnages for the hearing impaired trace
their ancestry to the French Sign Language, developed in the
mid-18th century by a priest, Father Charles Michel d'Epee.
A true manual sign language develops independently from
the langnage spoken around the deaf and, thus, has its own
word stock and grammar. So, deaf Parisians had a manual
sign langnage long before Father Michel learned it and, with
the aid of some of his pupils, devised signs for spelling out
words.
Thomas Gallaudet, founder of the first American school for
the hearing impaired, brought the French Sign Langnage to the
United States in 1816. Several sign langnages were current at
the time; Gallaudet combined these with the French system to
form the American Sign Langnage. However desirable, there
is no international sign langnage for the hearing impaired.
To express numbers, Roman numerals C for hundred and M
for thousand are brought into action.
lOO
To make "hundreds", the person signs the digit 1, 2, 3 ...,
followed by the sign for the letter C; to make "thousands", the
person makes the proper unit sign followed by the letter M - the
tips of three fingers of the right hand touching the left palm.
v x i !
2000

Chapter 2 SYSTEMS OF NUMERATION
Semaphores
The semaphore visual telegraph was invented by the French
CHAPPE Claude engineer Claude Chappe, who, with the backing of the Leg-
(1763-1805) islative Assembly of the French Revolution, built between
Paris and Lille (near the French-Austrian war front) a series
of towers on high places, spaced 10 to 16 kilometers apart.
Every tower was equipped with telescopes and a set of arm-like
structures - a semaphore - which could be turned around on
a pole; each arm had seven clearly discernible angular
positions, making a total of 49 combinations that were assigned
letters of the alphabet or other messages.
Greek, sema, "sign"; The word semaphore was a construction of Chappe's.
pherein, "to bear"
Semaphore telegraph towers were built also in other parts of
France, and Chappe's invention was soon copied elsewhere in
Europe.
Semaphore signaling with mechanically or electrically
controlled semaphore arms, often with rows of lights, is still
used to signal railroad trains.
Though currently at the point of extinction, semaphore
signaling between ships has been of great importance. A person
holds a small flag in each hand; with the arms extended, the
person moves the flags to different angles. Succeeding a
"numbers follow" signal, numbers from 1 to 9 are the same as
the first nine letters of the alphabet; zero is the same as the
tenth letter:
numbers A/1 B/2 C/3 D/4 E/5
follow
F/6
G/7
H/8 1/9 J/zero

69
Chapter
3
TYPES OF NUMBERS
Page
3.0 An Expanding Universe of Numbers 70
3.1 Rational Numbers 72
3.2 Prime Numbers 77
3.3 Perfect and Amicable Numbers 82
3.4 Irrational Numbers 84
3.5 Imaginary and Complex Numbers 87
3.6 The Quest for n 89
*** 7.5 Transfinite Numbers 257
*** 8.6 Figurate Numbers 289
Indicates cross-reference

Chapter 3 TYPES OF NUMBERS
An Expanding Universe of Numbers
<<(Do you know what the foundation of mathematics is?" I ask,.
"The foundation of mathematics is numbers, if anyone asked me
what ma/^es me truly happy, I would say: numbers. Snow and ice
and numbers. And do you know why?"
Peter H0eg, Smilla's Sense of Snow (1993)
(Die Zahlen sind freie Schopfungen des menschlichen Qeistes, sie
dienen als ein Mittel, um die Verschiedenfieit der Twinge leichter
und scfuirfer aufzufassen.
"Numbers are free creations of the human mind that serve as
a medium for the easier and clearer understanding of the
diversity of thought."
Richard Dedekind, Was sind und was sollen die Zahlen? (1887)
Starting with the natural numbers, or counting numbers,
1,2,3...
we have numbers where addition and multiplication will
always result in a natural number, but these numbers alone
are not sufficient for subtraction. Supplementing with zero
and negative whole numbers, thus obtaining the integers,
..., — o, — ^, — 1, 0, 1, Z, o, ... ,
we have enough numbers for addition, multiplication, and
subtraction of any integers but not for division, which
necessitates that we include numbers that are ratios of integers,
1 2 81
£•£•> o"> 7» 82* * ^e now nave integers and ratios of integers,
which all are known as rational numbers.
Rational numbers produce other rational numbers when
added, multiplied, subtracted, or divided. Yet rational
numbers are not always enough - no ratios of integers can
exactly represent the solutions of, for instance, x2 - 2 = 0, that
is, x = ± nor can numbers such as n and e be represented by
ratios of integers. Such numbers - not representable by ratios
of integers - are known as irrational numbers.
Rational numbers and irrational numbers are together
named real numbers.

Section 3.0 An Expanding Universe of Numbers
71
We can visualize the real numbers on the number line, a
straight line extending from negative infinity to positive
infinity and graduated in unit distances on both sides of the
origin, which symbolizes zero. Between the integers we have,
on the number line, all the remaining real numbers.
_5 _4 _3 _2 -1
0
Absolute Value
Ring
Field
The absolute value \a\ of a real number a is, by definition,
either positive or zero - it cannot be negative.
Real numbers cannot describe the solutions of an equation
such as x2 + 1 = 0, that is, x = ± V-l, for the product of any two
real numbers that have the same sign is always positive or
zero. Since real numbers cannot do the job, we take recourse to
complex numbers,
a + b\,
where a and b are real numbers and i is the imaginary unit,
defined
i2 = -1.
If a = 0, we have pure imaginary numbers, e.g.,
V^l = i; V-81 = 9 i,
and if b = 0, we have real numbers; thus, we may make the
whole "number universe" fit into the category of complex
numbers.
A number system in which addition, subtraction, and
multiplication are always defined and the associative and
distributive laws are valid is known as a ring; if division
(except by 0) can also be carried out, one speaks of a field.
We shall deal first with rational numbers and their properties,
before moving on to irrational numbers and, finally,
imaginary and complex numbers.
* * *
KRONECKER Leopold
(1823 -1891)
The German mathematician Leopold Kronecker claimed
that mathematical argumentation should be based only on
integers and finite procedures; from Kronecker originates
the dictum:
Qodmade the integers; all the rest is the worl^of Man.
We take issue with this much-treasured quote and argue
that the Ruler of Heaven and Earth might only have created
the counting numbers - and in the reverse order:
...,10 000 036,10 000 035,10 000 034,...,10191,10190,10189,...,
10,9, 8, 7, 6, 5, 4, 3, 2, 1
Big Bang i

72
Chapter 3 TYPES OF NUMBERS
3.1 Rational Numbers
Rational numbers derive their name from the fact that they
can all be written in the form of a ratio alb where the
numerator a may be any whole number and the denominator b
any positive whole number except zero.
The notation alb reads "a over 6" or "a divided by b".
If b is equal to unity, 6 = 1, the ratio alb is an integer = a; with
arbitrary values a and 6, we have a fraction.
Integers
Natural Numbers
All whole numbers, including negative numbers, zero, and
positive numbers, are called integers.
Positive integers are also known as natural numbers.
Natural Numbers
+
_5 _4 _3 _2 -1
0
i
1 J
I 1
I 1
I 1
I 1
1
3
4
i
Integers
Identity for Addition
Identity for Multiplication
BRAHMAGUPTA
(598 - after 665)
Zero was a long time in establishing itself as a number in its
own right; it was hard for mathematicians to accept that
"nothing" could be regarded as "something".
In one respect, zero is not equal in status to other numbers:
division by zero is not permitted.
Zero is the identity for addition (and subtraction) of numbers,
since x + 0 = 0 + x=x for all x. One is the identity for
multiplication of numbers, since x • 1 = 1 • x = x for all x.
Negative numbers were also long denied legitimacy in
mathematics. We have no evidence of negative numbers
being recognized in Babylonian, Pharaonic, ancient Greek,
or any other ancient civilization. On the contrary, the Greeks
considered geometry the only acceptable form of mathematics,
and since distance cannot be negative, they had no use for
negative numbers.
In the 7th century, negative numbers were used in
bookkeeping in India; positive quantities denoted assets, negative
ones debts. The Hindu astronomer Brahmagupta, in a chapter
dealing with mathematics in his work on astronomy
from about A.D. 630, shows a clear understanding of negative
numbers.

Section 3.1 Rational Numbers
73
CARDANO Girolamo
(1501 -1576)
The earliest documented evidence of the use of negative
numbers in European mathematics is the Ars magna, published in
1545 by the Italian mathematician Girolamo Cardano, who,
true to Renaissance custom, engaged in many other sciences
as well; as a physician, he gave an early clinical description
of typhoid fever.
von LEIBNIZ Gottfried Wilhelm
(1646 - 1716)
HltRONYMI
CAR DAN I PRA.
STATnTT ISSIM1 MAT HE ,
matiuct Fhilofophi, ac Medici Pvii.
ARTIS MAGNAE,SIVE
DEREGVHSALGE3R.AICIS LI B VNiu
tut ct tatiut Oj*tit dcA.ztthtn£tici.yttoJL OF VSZZ1L.
rECTVM.iufcrijfif.eft in.decline Deccmilf.
In the early 17th century, mathematicians began explicitly
to use "negative numbers" but met with heavy opposition.
Descartes called negative roots "false roots", and Pascal was
convinced that numbers "less than zero" could not exist.
Leibniz admitted that negative numbers could lead to absurd
conclusions and misconceptions, but defended them as useful
aids in calculation.
The general acceptance and algebraic use of negative
numbers came during the 18th century, although there were still
mathematicians who did not feel at home with them and quite
often tried to avoid using them.
- In some situations we still have not fully adopted the
minus sign and the idea of negative numbers: The
modern corporate report gives losses as (138 000), rather
than a gain of -138 000.
To facilitate the reading of numbers with many digits, these
are commonly divided into groups of three, beginning on the
right and working toward the left. The separation is often
done by means of a small space rather than by a comma, a dot,
or any other mark that might cause misunderstanding:
123 456 789
1234 567
Four-digit numbers need not be divided: 5367
In India, traditionally, long numbers are divided so that the
terminal three digits form one group and the remaining digits
form groups of two or, for the initial group, only one. Thus, the
number of the approximate 900 million inhabitants of India
would be written 90, 00, 00, 000.

Chapter 3 TYPES OF NUMBERS
OU-Timers* Counting
As a sample of reading large numbers in your great-great-
great- ... and a few more greats ... -great-grandparents' time,
let us take an excerpt from the Bamberger Rechenbuch of 1483:
J(l fc<b*vnb a<foig taufcnt taufcnt maltan
fcnt/fibcnpunbccttaufcnt maltaufcnt /ntxirt
vnnb a$%i$ tauftntmaitaufcnt /bmjmnfccre
tau(tnt/\iin\fvn$wcnqi$tauf<nt/tinj)Unbctt
vnba<btvnb(il>cn%i$.
which tells us how to read out the number
86 789 325 178;
like this:
six-and-eighty thousand thousand times thousand,
seven hundred thousand times thousand,
nine-and-eighty thousand times thousand,
three hundred thousand,
five-and-twenty thousand,
one hundred and eight-and-seventy.
Taken in installments, it isn't so very difficult, is it now?
Fractions
Common fractions have the form 7- or a lb. where the
6
numerator a may be any integer and the denominator b any
integer > 0; the notation reads "a over 6" or "a divided by 6".
If the absolute value of the numerator of a common fraction
alb is smaller than the denominator, \a\ < b, we speak of a
proper fraction; if it is not, \a\ > 6, we get an improper
fraction, which can always be split into an integer and a
proper fraction,
8 2 5 2
33' 33*
All common fractions have exactly defined positions on the
number line:
_8 1 5 18
3 2 2 5
-5-4-3-2-1012345
There is always another rational number between any two
rational numbers p and q; their arithmetic mean, ^(p +0.), is
placed halfway between p and q on the number line. This
interpolation can be repeated any number of times, so the
number of rational numbers is infinite.
The sum, difference, product, and quotient of two fractions are
always another fraction, or an integer.

Section 3.1 Rational Numbers
75
Decimal Fractions
Decimal fractions are numbers that consist of an integer part
- which may be zero - and a decimal part less than unity
that follows the decimal marker, which may be a point or a
comma.
An unfortunate practice has developed in English-speaking
countries of curtailing the expression "decimal fraction" and
calling the entire number by the name of "decimal", a word
that should properly be reserved for any of the digits following
the decimal marker - a practice respected by all other
countries.
In this text, we shall call the complete number a decimal
fraction, the digits following the integral part to be known as
decimals.
Examples of decimal fractions are 0.90, 3.02, and 4.005, which
read out "zero point ninety" or "oh point nine oh", "three point
oh two", and "four point double-oh five", respectively; point is
short for decimal point.
The SI - the International System of Units - recommends
the use of a decimal comma on the line as decimal marker
- 0,90; 3,02; 4,005 - and this practice is followed by most
countries: in general, those countries that did use a comma
already before the SI recommendation.
There is nothing to suggest that the "decimal-point" nations
- among them the United States and English-speaking
countries in general - will soon change from the decimal point to
the comma. Even great institutions and large companies of
world renown which do otherwise swear by SI still use the
decimal point.
In the British press the decimal marker is sometimes a dot
raised halfway above the line - a disturbing practice as it may
be confused with a multiplication sign.
In this text, we use the decimal point, on the line.
To avoid missing the decimal point, decimal fractions whose
absolute value is less than unity should be written showing the
units zero in front of the decimal point; thus: 0.93 .
In long decimal fractions, digits are usually divided into
groups of three, working both ways from the decimal marker:
12 346 582.006 852 3...
In scientific tables, however, the digits are often in groups of
five.

Chapter 3 TYPES OF NUMBERS
Conversion of Fractions
Every common fraction can be converted to a decimal fraction
by carrying out the division implied by the fraction bar, with a
decimal marker separating the integer part of the number
from the decimal part.
The decimal part can be looked upon as a series of common
fractions with denominators ten, a hundred, a thousand, etc.;
for instance:
5 6 6 6
-= 1.666... = 1 + — + rrrr +
3 10 100 1000 **•
When converting common fractions to decimal form, we
distinguish two cases.
Common fractions such as 2/5, 3/4, and 7/8 give decimal
fractions 0.4, 0.75, and 0.875, respectively, that is, finite or
terminating decimal fractions, whose sequences of decimals
have a definite break-off point after which all the places are
zeros.
Other fractions will produce endless sequences of decimals:
5/6 = 0.833 333 333...
1/96 = 0.050 505 050...
1/27 = 0.037 037 037...
1/74 = 0.013 513 513...
We observe that the decimals repeat periodically, that is,
we have periodic nonterminating (or infinite) decimal
fractions. The periods are, in the above cases, 3; 05; 037; and
135; they may be much longer when the denominator of the
common fraction is a prime number or a large number.
The fraction 1/29 gives a decimal number with a 28-digit
cycle; 1/211 has a period of 45 digits.
The periodicity may be marked by placing a bar above the
repeating cycle; thus:
5/ 6 = 0.83
1/27 = 0.037
1/74 = 0.0135
1/29 = 0.034 482 758 620 689 655172 413 7931
To convert the periodic decimal expression 0.0135 to a common
fraction, we write:
1000 x = 13.5135135 135...
- x = - 0.0135135135...
999 x = 13.5
135 5-9-3 1
x =
9990 9-3-74-5 74

77
3.2 Prime Numbers
FIBONACCI Leonardo
(c. 1170 - c. 1250)
Sieve of Eratosthenes
ERATOSTHENES
(c. 276 -194 B.C.)
Greek mathematician,
astronomer, and geographer
Most integers are composite numbers, that is, they can be
written as products of two or more integers, each larger than 1,
eg-,
120 = 2x2x2x3x5
462 = 2x3x7x11
16 873 = 47x359
An integer larger than 1 that is divisible only by 1 and itself is
called a prime number, or a prime. Leonardo Fibonacci of
Pisa, in 1202, used the name incomposite numbers, as opposed
to composite numbers.
The simplest means of sifting out prime numbers from
natural numbers is the sieve of Eratosthenes. In a sequence of
natural numbers, beginning with 2, start by striking out every
second number after 2 - this eliminates all multiples of 2.
Next delete every third number after 3, then every fourth
number after 4, etc. Several numbers will be canceled more
than once.
The numbers left standing are the prime numbers; those
struck out are composite.
Thus, the sequence of primes begins
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37 ...
and goes on indefinitely,
...1000 000 009 649, 1000 000 009 651 ....
Primeness is a property of the number itself; e.g., fifty-nine is
prime whether written 59, binary 111011, or in any other
system of numeration.
If a prime divides the product of two integers, it divides at least
one of the factors.

78
Chapter 3 TYPES OF NUMBERS
EVCLEIDIS Proof: p. 159
(3rd century B.C.)
Fundamental Theorem
of Arithmetic
The first person known to have proved that the number of
primes is infinite was Euclid, who called them "numbers
measured by no number but a unit alone", where "measured"
means divisible by a given number.
The fundamental theorem of arithmetic states:
Every positive integer greater than 1 is a prime or can
be expressed as a unique product of primes and powers of
primes.
For instance, with the primes in ascending order, 360 has the
unique product of powers of primes
23 • 32 • 51.
Fermat Numbers
deFERMAT Pierre
(1601 -1665)
EULER Leonhard
(1707 -1783)
Prime numbers have always intrigued mathematicians, who
want to know how and if they might be predicted. Many
supposedly prime-generating formulas have been suggested, but
they have all failed in testing, and mathematicians are
beginning to doubt that there will ever be a practical, simple
formula guaranteed to produce exclusively prime numbers.
Fermat numbers, named after Pierre de Fermat, a French
lawyer and gentleman scholar, are described by the formula
Fp = 2^ + 1.
Knowing that p values from 0 to 4, inclusive, did produce
primes,
*b = 3
Fi = 5
F2 = 17
Fs = 257
F4 = 65 537,
Fermat in 1640, somewhat rashly, prophesied that all
numbers of this form would be primes, for all positive integers p
- but he was proved wrong.
In 1732, Leonhard Euler proved that the fifth Fermat number,
/*5, can be factorized,
F5 = 4294 967 297 = 641x6 700 417,
and, consequently, is not a prime but a composite number.
In 1880 it was shown that F6 has a prime factor 274 177; in 1905,
Fj was also proved to be composite, and in 1971 it was factorized
into two prime factors of 17 and 22 digits, respectively.
Fg, with 78 digits, factorizes into two prime numbers, one with
16 digits and the other with 62 digits.
In 1992, F$ was also factorized, and it is known that Fiq to F21,
inclusive, and some other Fermat numbers are composite.

Section 3.2 Prime Numbers
79
The belief is growing that all Fermat numbers greater than F4
are composite numbers.
All numbers of the form
Fm = 2™ + l,
where m is not a power of 2, are composite numbers.
MERSENNE Marin
(1558 -1648)
French mathematician
and Franciscan friar
LUCAS
Frangois-Edouard-Anatole
(1842 -1891)
French number theorist
391581
Mersenne Numbers and Mersenne Primes
Fermat had communicated his mistaken belief that all
Fermat numbers would be primes to Marin Mersenne who, in
his work Cogita physico-mathematica (1644), studied numbers
of the form
Mp= 2P-1,
where p is a prime number.
Mersenne was of the opinion that numbers of this kind - now
known as Mersenne numbers - would be primes for the
following prime values of p,
p = 2, 3, 5, 7, 13, 17, 19, 31, 67, 127, and 257;
and composite numbers for all other/? smaller than 257.
It is not known on what grounds he made this assumption, but
it was very nearly correct. It could not be properly
substantiated till the advent of the electronic computer in 1947 when,
against Mersenne's belief, it was shown that Mqj and M257 are
composite numbers, and that Mgi, Mg9, and M107 joined the
ranks of Mersenne primes. The largest number proved to be
prime, without the help of the electronic computer, is M127
(Lucas, 1876).
If p is not a prime number, Mersenne numbers are always
composite numbers.
The rapid increase of the function 2P causes Mersenne
numbers to grow at a fascinating rate, and to lend themselves
admirably to the generation of very large prime numbers.
In 1994 the known and proven Mersenne primes numbered 32;
the most recently found are:
Prime
^132 049
^216 091
2 216193 _i
^756 839
^859 433
^i 257 787
^2 976 221
Number
of Digits
39 751
65 050
65 087
227 832
258 716
378 632
895 932
Discovered by
David Slowinski
David Slowinski
Noll et al.
David Slowinski
David Slowinski and Paul Gage
David Slowinski and Paul Gage
Gordon Spence
Year
1983
1983
1989
1992
1994
1996
1997
The prime 391581 • 2216193 - 1 was found by means of an
algorithm not entirely based on Mersenne numbers, and is not
a Mersenne number.
The 895 932 digits of the latest found prime would fill up 450
pages of a paperback book in standard type.

80
Chapter 3 TYPES OF NUMBERS
Prime Testing Method of Fermat
Pseudo-Primes
Fermat has given us a method of finding out if a specific
number p is prime or not: If p is a prime number and a is an
arbitrary number smaller than p, then
oP-1-!
will be divisible by p.
Thus, if it is not, p cannot be a prime.
If 2P " 1 - 1 is divisible byp, that does not mean, however, that/?
must be a prime, as there may exist composite numbers with
the same property, e.g., 341 = 11 • 31; it can be proved that 2340 -
1 is divisible by 341.
Such composite numbers are called pseudo-primes.
Thus, if 2P ~ 1 - 1 is divisible by p, we can only say that p is
either a prime or a pseudo-prime, and more likely the former.
Of the first 1000 natural numbers, only three - 341, 561, and 645
- are pseudo-primes; of the first million, only 245.
The Prime Number Theorem
LEGENDRE Adrien-Marie
(1752 -1833)
Natural Logarithms: p. 155
GAUSS Carl Friedrich
(1777 -1855)
Integrals: pp. 721 et seq.
HADAMARD Jacques Salomon
(1865 -1963)
de la VALLEE-POUSSIN C.J.
(1866 -1962)
The number of primes is infinite, but they occur less
frequently as one goes farther out in the number sequence. Since
the time of Euclid, mathematicians have tried to formulate a
law, the prime number theorem, relating the number - n(n) -
of primes lower than, or equal to, a given integer n, to n. Here
n has nothing to do with the number n9 but is used to indicate
prime ("^rime,>).
Adrien-Marie Legendre in 1778 published his work Essai sur
la theorie des nombres, where he suggested a modified form of
the first approximation, n(n) & n/ln n,
n(ri) &
n
In n - 1.08366 '
where In n is the natural logarithm of n, and 1.08366 is an
empirically found correction term.
Carl Friedrich Gauss's discovery - the integral logarithm -
was published in 1863, eight years after his death,
n{n) & hi(n) =
n
r dx
j
2
In n
These conjectures had to wait for final proof until 1896 when
the Frenchman Jacques Hadamard and the Belgian Jean de
la Vallee-Poussin independently proved that the Legendre
formula is the better approximation of n (n) for n values up to a
million, the integral logarithm taking over for very large n.

Section 3.2 Prime Numbers
81
n
n
103
104
105
106
107
108
K \Tl)
168
1229
9592
78 498
664 579
5 761455
In n
- 1.08366
172
1231
9588
78 534
665138
5768 004
±ji yn)
178
1246
9630
78 628
664 918
5762 209
GOLDBACH Christian
(1690-1764)
Born in Konigsberg, Prussia;
professor of mathematics
and historian of the Imperial
Academy at St. Petersburg
VINOGRADOV Ivan M.
(1891 -1983)
Goldbach Conjectures
Prime numbers exhibit many unexpected and unexplained
properties:
In a letter to Leonhard Euler in 1742, Christian Goldbach
conjectured that every even integer greater than 2 can be written
as the sum of two primes:
4 = 2 + 2 8 = 5 + 3 12 = 7 + 5 16=11 + 5
6 = 3 + 3 10 = 7 + 3 14=11 + 3 18 = 11 + 7 etc.
No exception to this conjecture is known; nor is a valid proof.
Another conjecture of Goldbach's maintains that every
sufficiently large integer number - 6 or higher - can be written
as a sum of three primes:
6 = 2 + 2 + 2 8 = 2 + 3 + 3 10 = 2 + 3 + 5
7 = 2 + 2 + 3 9 = 3 + 3 + 3 11 = 3 + 3 + 5 etc.
Also for this conjecture, no exception is known, nor is there a
valid proof.
It is known that every sufficiently large odd number can be
expressed as the sum of three primes (Vinogradov's theorem,
1937).
Twin Primes
Many primes appear in pairs as twin primes that differ by 2:
3, 5 - 11, 13 - 17, 19 - 29, 31 - 41, 43 ...
Experimental evidence supports the infinitude of such pairs,
but a strict proof is lacking.
Applications
Prime numbers and methods of factoring large composite
numbers play a leading part in today's methods of concealing
data from illegitimate access.
In speed-reduction gears, it is often advantageous to use gear
wheels whose numbers of teeth are prime numbers - or any
other numbers whose only common factor is 1. The same gear
teeth will then mesh only at long intervals, limiting
uncontrolled localized wear of gear flanks by high spots and other
imperfections, and reducing gear noise.

82
Chapter 3 TYPES OF NUMBERS
3.3 Perfect and Amicable Numbers
Unlike prime numbers, composite numbers can be broken
down into factors. Some composite numbers have attracted the
attention of reputable mathematicians for serious
investigations into number theory and have inevitably caught the fancy
of various mystics, soothsayers, faith healers, and other such
people to further their dubious pursuits.
Among these groups of composite numbers are perfect numbers
and amicable numbers, or friendly numbers.
Perfect Numbers
Perfect Numbers
NICOMACHOS
(c. A.D. 100)
EULER Leonhard
(1707 -1783)
An integer number that is equal to the sum of all its possible
divisors - except the number itself - is called a perfect
number. If the sum is less than the number itself, the number
is said to be defective or deficient; if greater, the number is
abundant.
The lowest perfect numbers are 6, 28, 496, and 8128.
6 is divisible 28 is divisible 496 is divisible
by 1, 2, 4, 7,
and 14:
by 1, 2, 3:
by 1, 2, 4, 8, 16,
31, 62, 124, 248:
1 + 2 + 3 = 6
1 + 2 + 4 + 7 + 14
= 28
1 + 2 + 4 + 8 + 16
+ 31 + 62 + 124
+ 248 = 496
Euclid proved in his Elements (Book IX), that if (2P - 1) is what
we now designate as a Mersenne prime, then
N = 2P~1(2P-1)
is a perfect number.
Nicomachos of Gerasa suggested that the converse of the above
theorem was true, that is, that the formula gives all the perfect
numbers without exception. Euler proved in 1750, a partial
converse of that theorem: If N is an even number perfect
number, then it has the form established by Euclid, N = 2P~l
(2P - 1). The full converse has not been proved as yet.
All perfect numbers known today - about 30 - are even. There
is reason to believe that no odd perfect numbers exist; at all
events, it has been proven that there is none smaller than
10100 (= 1 googol). The hunt for an odd perfect number or the
proof that there is none goes on as of 1997.
Every perfect number except 6 can be written as a sum of the
cubes of an unbroken sequence of consecutive odd integers:
28 = l3 + 33
496 = l3 + 33 + 53 + 73
8128 = l3 + 33 + 53 + 73 + 93 + ll3 + 133 + 153; etc.
The single-digit cross sum of every perfect number, except 6,
is 1:

Section 3.3 Perfect and Amicable Numbers
83
28 =>2 + 8 = 10 =>l + 0 = 1
496 => 4 + 9 + 6 = 19=>1 + 9=>10=>1 + 0 = 1
8128 =>8 + l + 2 + 8 = 19 => 1 + 9 => 10 => 1 + 0 = 1; etc.
The sum of the inverses of the factors of a perfect number,
leaving out the unity factor but including the number itself, is
also 1:
6 ^2 +3 +6 = 1
oq 1111 i,
28=>2+4+7+14 + 28 _1
i i i i i i 1 J L_i
496 => 2 + 4 + g + 16 + 31 + 62 + 124 + 24g+ 496 - 1
etc.
The table of possible perfect numbers begins:
n
1
2
p
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
2P'1
1
2
4
8
16
32
64
128
256
512
1024
2048
4096
8192
16 384
32 768
2^-1
1
3
7
15
31
63
127
255
511
1023
2047
4095
8191
16 383
32 767
65 535
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
prime
prime
3x5
prime
3x3x7
prime
3x5x17
7x73
3x11x31
23x89
3x3x3x5x91
prime
3x 43x 127
7x 31x 151
3x5xl7x 257
N
-
6
28
-
496
-
8128
-
-
-
-
-
33 550 336
-
-
_
Amicable Numbers
Two integer numbers are said to be amicable, or friendly, if
each is the sum of all the possible divisors of the other; the
smallest pair of friendly numbers is 220 and 284:
- 220 is divisible by 1, 2, 4, 5, 10, 11, 20,22, 44, 55, and 110,
which add up to 284;
- 284 is divisible by 1, 2, 4, 71, and 142, whose sum is 220.
The pair 220, 284 was long the only known, and generally
thought to be unique, until
in 1636, Pierre de Fermat found the pair 17 296 and 18 416; and
in 1638, Rene Descartes discovered the pair 9 363 584,9 437 056.
In 1750, Leonhard Euler added a further 60 pairs of amicable
numbers to the total; and in 1866, the pair 1184,1210, until then
overlooked, was found by the 16-year-old Nicolo Paganini, not
to be confused with his violin virtuoso namesake (1782 -, 1840).
The number of pairs of amicable numbers known today
exceeds 1000.

Chapter 3 TYPES OF NUMBERS
Irrational Numbers
Irrational numbers are numbers that are not rational in the
sense that they cannot be expressed as common fractions, that
is, as a ratio of two integer numbers.
Writing an irrational number in decimal form will produce
an endless sequence of decimal digits - a nonperiodic nonter-
minating decimal number.
Irrational numbers are either algebraic numbers or
transcendental.
Algebraic Numbers
Algebraic numbers are roots of algebraic equations with
integer coefficients; such equations have either rational roots
4a2-9 = 0; a = ±3/2
or irrational solutions; the roots of the equation
a2-2 = 0
are the "square roots of 2", a = ± *v/2 .
In the sixth century B.C., the Pythagoreans - an ancient Greek
philosophical school and religious brotherhood - encountered
the number y2 (in our notation) as the length of the diagonal of
a square whose sides are one unit in length. They lived by the
dictum that all is number, believing that all things could be
explained as relationships between numbers, which to them
meant positive integers and common fractions.
This "number" did not fit into the ideas of the Pythagoreans,
and they labeled it an anomaly of the square - only to find that
the world was full of such anomalies, for instance, the square
root of three,
V3 = 1.732 050 807...,
and the cube root of two,
V2 = 1.259 92105....
Now V2 lies between 1.41 and 1.42, because 1.412 = 1.9881 and
1.422 = 2.0164; further calculation places V2 between 1.414 and
1.415 and closer to 1.414, because 1.4142 = 1.999 396 and 1.4152 =
2.002 225 . Continued calculation with an electronic calculator
will give values with an increasing accuracy,
V2 = 1.414 213 562....

Section 3.4 Irrational Numbers
85
Transcendental Numbers
LIOUVILLE Joseph
(1809-1882)
French mathematician
Transcendental numbers are not the roots of any algebraic
equation. The existence of transcendental numbers was
proven in 1844 by Joseph Liouville.
Transcendental numbers do, in fact, make up the vast
majority of all numbers.
Two transcendental numbers command special interest:
e, the base of natural logarithms, and
7i, the ratio of the circumference of a circle to its diameter.
o
o
p. 774
EULER Leonhard
(1707 -1783)
HERMITE Charles
(1822-1901)
French mathematician
( 1
is the limit value of the expression 1 + — I raised to the ra-th
V
n
power, when n increases indefinitely,
e = lim
n —> °°
V n
Jl
= 2.7 1828 1828 45 90 45....
The decimals have been grouped so as to enhance the
peculiarity of the recurring groups 1828 and the two 45's and the 90
of the first fifteen decimals; these groups of digits may serve
as a mnemonic.
The symbol e was first used by the Swiss mathematician
Leonhard Euler in accounts of his results, in letters written in
1727 or 1728 from St. Petersburg, and again in 1731. In print, e
appeared in 1736 in his Mechanica, possibly inspired by the
word exponential; today, it is generally regarded as a homage
to Euler.
Euler showed, in 1737, that e is irrational; in 1873 Charles
Hermite added the proof that e is transcendental.
JONES William
(1675 -1749)
71 is the symbol used today to represent the ratio of the
circumference of a circle to its diameter.
This designation was not introduced until 1706, when used by
William Jones in his Synopsis palmariorum matheseos,
probably after the initial letter of the Greek rcepi^epeicc, "periphery".
Until then, instead of n9 one had to content oneself with the
quaint Latin phrase
"quantitas, in quam cum multiplicetur diameter, provenient
circumferentia",
meaning "the quantity which, when the diameter is multiplied
by it, gives the circumference".

86
Chapter 3 TYPES OF NUMBERS
EULER Leonhard
(1707-1783)
It is due to the great prestige of the Swiss mathematician
Leonhard Euler that we use n with today's meaning. In his
early writings, Euler had frequently used p to denote the
circumference-to-diameter ratio, but changed to n in his
textbook Mechanica, published in 1736.
MECHANICA
SIVE
MOTVS
SCIENTIA
ANALYTICE
EXPOSITA
AVCTORE
LEONHARDO EVLERO
ACADEMIAE IMPER. SC1ENT1ARVM MEMBRO ET
MATHESEOS SVBLIMIORIS PROFESSORE.
TOMVS I.
1NSTAK SYTPLEMEKTI AD COMMEHTAR.
ACAD. SCIENT. IMPER.
PETROPOLI
EX TYPOGRAPHIA ACADEMIAE SCIENTIARVM.
a. natf.
PROPOSITIO 20.
Theorems.
154. Congruente puncii divectione motus cum
pttmtiae direction^ erit incrementum ecleritatis, vt po*
Saiia'dufta in tempuscuhm et diuifa per material* feu
Quahtitatem pun3u
Demonflratio.
Sine duo puncca feu corpuscula inaequalia A et
Bmota in reftis AM, BN. Sollicitentur ea a po-
tentiis^ et n refpectiue, dum percurrunt fpatiohi
Jttax, Nfl, et fint tempora, quibus ea percurrun-
tur dt, dr., Mahifeftum eft pun<ftum B a potentia tt
codem modo afnei ac puncflum A a potentia-5-
(13^.)- Quare fubfticuto loco B puncfto ipfi A
jiequali, pro potentia tz fubftitui debet potentia if?
hocque modo obtinemus cafum propofitionis prae-
cedentis, quo punefca pbnuntur aequalia. Hanc ob
fpvoi.incrementum celeriratis per Mm eft ad incrc-
incntam ecleritatis per N>/ vt pdt ad -r^r, feu vc
^"ad-j-(150-). .Ex quo conftat propoficum, quod
ecleritatis incrementum, fie vc fadhim ex potentia
ct^tempusculo diuifum per pun&i materiam feu
quantitatem. Q. E. D.
von LINDEMANN
Carl Louis Ferdinand
(1852-1939)
In Euler's time, nothing definite was as yet known about the
nature of n, although mathematicians in general agreed that e
and n were probably not roots of algebraic equations.
The German mathematician Ferdinand von Lindemann, in
1882, succeeded in proving that n is transcendental. The area
of a circle is radius2 • n, that of a square, side2; consequently,
the side of a square whose area is equal to that of a circle with
radius 1 is ^fk. A construction with straightedge and compass
alone can give only lengths that are algebraic numbers, so
Lindemann's proof that n is transcendental was conclusive
evidence that the age-old problem of squaring the circle is
unsolvable.
The transcendentality of en was proved in 1929, that of 2^2 in
1930. Yet, even if we know today that the number of
transcendental numbers is infinite, there are still many irrational
numbers, e.g., nn, that defy our curiosity whether they are
algebraic or transcendental.

87
3.5 Imaginary and Complex Numbers
The Imaginary Unit
Imaginary and complex numbers were long looked upon with
suspicion, and their possible existence was politely ignored.
Cardano called negative numbers "fictitious" and elevated
imaginary numbers to "sophistic". The Italian
mathematician Raffaele Bombelli introduced imaginary and complex
numbers in his treatise L'Algebra (1572) in connection with
the solving of cubic equations.
Negative numbers have no square roots that can be expressed
by real numbers. To overcome this problem, the imaginary
unit has been introduced, defined as the square root of minus
one,
i = V-l ; i2 = -1.
Imaginary numbers have proved a very useful tool in a great
many scientific and practical applications. To the scientist,
imaginary numbers are indispensable and just as "real" as
real numbers themselves.
The imaginary unit is generally represented by the symbol
i - for imaginary - except in electrical engineering where j
is used to avoid confusion with the symbol i, which denotes the
instantaneous value of an electric current.
Complex Numbers
Complex numbers take the form
z = a + bi,
where a = Re(z) is the real part and b i = Im(z) the imaginary
part of z.
Besides z, the letter w is often used to denote complex numbers.
Imaginary numbers, like real numbers, can be presented on a
straight number line. This is not possible, however, with
complex numbers, which require two independent coordinates
for their representation. Thus, by bringing in geometry, the
imaginary unit and complex numbers become just as "real"
as real numbers themselves.
The Complex Number Plane
The idea of representing complex numbers with points in a
coordinate plane originated with the English mathematician
John Wallis in his De algebra tractatus (1685), but it was put
forward in a rather vague manner and had no influence on
contemporary mathematics.
CARDANO Girolamo
(1501 -1576)
BOMBELLI Raffaele
(1526 -1572)
WALLIS John
(1616 -1703)

88
Chapter 3 TYPES OF NUMBERS
WESSEL Caspar
(1745 -1818)
ARGAND Jean Robert
(1768-1822)
GAUSS Carl Friedrich
(1777-1855)
The first practical representation of complex numbers dates
from 1797 when the Norwegian cartographer Caspar Wessel
read a paper, "Om Directionens analytiske Betegning ..."
("On the Analytic Representation of Direction"), before the
Danish Academy of Sciences, published the following year
in the Memoires of the Academy. WessePs work went
essentially unnoticed, however, until a French translation appeared
a century later, in 1897.
In 1806, the Geneva-born Parisian bookkeeper Jean Robert
Argand published, anonymously, in a small privately printed
edition, a method of representing imaginary numbers
geometrically. This work might have suffered the same fate as
Wessel's paper, except for J. F. Frangais, a professor of
military engineering, who found a copy of it and invited the author
to come forward and acknowledge his work.
As often happens when "the time is ripe", Wessel and Argand
had published their accounts of essentially the same idea
independently and simultaneously; neither Wessel nor Argand
explicitly mentioned a complex number plane, a name we owe
to the German mathematician Carl Friedrich Gauss.
The usual form of representing a complex number z = x + y i
graphically is by presenting its real part x along the horizontal
real axis and its imaginary part y on the vertical imaginary
axis of a Cartesian coordinate system; the complex number
will then appear as a point (x, y) in the coordinate plane.
The graph illustrates the positions of two complex numbers,
Pl = 4 + 3i
P3= -2-5i
ESSAI
SUa UNE MANIERE
SE REPKBStNTER
LES QU ANTITESIMAGINAIRES
DANS LES CONSTRUCTIONS
G£OMJ£t&IQUES.
A PARIS
M.DCCC.VL
Title page of Argand's work on geometric
representation of imaginary numbers. From
an 1874 facsimile (Gauthier-Villars, Paris).

89
The Quest for n
The circle and everything pertaining to it have held captive
humankind's interest, sometimes amounting to fascination,
for several thousand years - far longer than any other single
feature in mathematics.
People of all kinds - professional mathematicians, amateurs,
and dilettantes - have felt the desire "to square the circle", that
is, to construct a square with the same area as a given circle.
To do this, one must know the ratio of a circle's circumference
to its diameter. Since the time of Leonhard Euler, we denote
this ratio by the Greek letter n.
The Ancients
Nearly all peoples of the ancient world used the number 3 for
the ratio of a circle's circumference to its diameter as an
approximation sufficient for everyday needs. In cases where
a higher accuracy was needed, e.g., when matters of law
might be involved, our ancestors developed what they
considered "more exact" values.
From a Babylonian tablet of sunbaked clay found in 1936 at
Susa it appears that, besides 3, the Babylonians used the value
n = 3g = 3.125 .
In the Rhind Papyrus of old Egypt we find a solved problem
that states that the area of a circle of nine length units in
diameter is the same as the area of a square whose side is eight
units of length, which gives
7i-92 o 256
—-— = 82; n = — = 3.160 49 ....
4 81
The early Greeks also began with a n = 3 for everyday use; for
matters more serious they evolved other values which were not
much better, e.g.,
n = VlO = 3.1622....
These values of n were all empirical in the sense that they were
based exclusively on practical experience, not on theoretical
considerations.
The most important contribution of the early Greeks to our
knowledge of n is the "method of exhaustion" attributed
variously to Antiphon, to Euclid, and, more likely, to Eudoxus.
The perimeter of a regular polygon with n sides - an n-gon -
inscribed in a circle is, of course, shorter than the
circumference of the circle; the perimeter of a circumscribed regular
n-gon is longer.

90
Chapter 3 TYPES OF NUMBERS
HIPPARCHOS
(c. 147 - after 127 B.C.)
ARCHIMEDES
(c. 287 - 212 B.C.)
If n is made sufficiently large, the perimeters of the two
polygons can be made to approach the periphery of the circle
arbitrarily close - one from within, and one from without -
and enclose it within ever narrower limits until the areas
between each of the polygons and the circle periphery are
"exhausted".
In the 2nd century B.C., Hipparchus computed an extensive
table of chords and proposed the value
n =
377
120
= 3.14166,
which is correct to four decimal places, a remarkably good
approximation for his time.
Archimedes
Archimedes of Syracuse, regarded as the greatest scientist-
mathematician of antiquity, applied his method for
calculation of arc length to determine n.
Beginning with regular hexagons - inscribed in, and
circumscribed to a circle - and doubling the number of sides
four times until he had a pair of regular 96-gons, he calculated
the lengths of the perimeters of the successive polygons,
^6 P6 Pl2 P12 ?2A P24 ^48 P48 ^96 P96 >
where Pn and pn denote the perimeters of the circumscribed
and inscribed ra-gons, respectively, by the recursion formulas,
ZpnPn
?2n =
P2
n = vPnP2n »
Pn+Pn *
that is, by taking alternately the harmonic and the geometric
means.
For the 96-gons, Archimedes calculated the approximate limits
for 7i,
0 10 _ 10 1
371<7C < 370 =37 '
or, in decimal notation,
3.1408... < n < 3.1428....
These values appear in the extant part of Archimedes' book On
the Measurement of the Circle as
"The ratio of the circumference of any circle to its
diameter is less than 3 1/7 but greater than 3 10/7i. "
After Archimedes, no essentially new ideas for the
calculation of n were suggested until the development of calculus
toward the end of the 17th century.
A Widened Outlook
While no scientific progress took place in Europe during the
Dark Ages - the thousand years starting with the fall of the
West Roman Empire in 476 - mathematics and other sciences

Section 3.6 The Quest for n
91
FIBONACCI
Leonardo of Pisa
(c. 1180-1250)
al-KASHI (or al-KASHANI)
Jamshid Masud
(? -1429)
progressed, albeit very slowly, in countries outside Europe
where the Church had no influence.
In India, as in most countries of the day, astronomy was the
foremost science, with mathematics considered only as an
adjunct to it.
Aryabatha, in 499, published n = 3.1416...,
Brahmagupta, born in 598, used n = VlO = 3.162 2...,
Bhaskara, born in 1114, held that n = 3.141 56...,
all of them without telling why and wherefor.
In China, Liu Hui in A.D. 263 published the limits
3.141024... < n < 3.142 764...
obtained for a pair of 96-gons; for a 3072-gon, he found
n = 3.14159....
An interesting n value was suggested by the astronomer
Tsu Chung-chih (430 - 501),
355
n = — = 3.1415929...,
which is correct to six decimal places, and was not to be
bettered in Europe until the 16th century, more than a thousand
years later.
In Europe, around the year 1000, a new dawn began slowly to
rise in the universities where the thirst for knowledge could no
longer be denied.
When in 1085 Toledo was recaptured from the Moors, the
authorities did not order the destruction of its library. Instead,
scholars from all Europe were invited to come to Toledo and
assist in translating the invaluable manuscripts of scientific
books into Latin from the original Greek or from Arabic or
Hebrew translations.
The first great mathematician of the Western world, marking
the end of the Middle Ages, was Fibonacci, or Leonardo of
Pisa, who traveled extensively in Arab countries and
developed an admiration for their knowledge, which he brought
back to Italy.
Like Archimedes, Fibonacci studied the 96-gon, from which
he calculated the value - published in 1220 in Practica
geometriae -
864
n = ==- = 3.141818...
which is correct to three decimal places.
In Persia, Jamshid Masud al-Kashi in 1424 published Risala
al-muhitiyya ("Treatise on the Circumference") with the
result of his calculations for an inscribed 3 • 228-gon,
n = 3.141592 653 589 793 25...
which is correct to sixteen decimal places, thereby surpassing
all earlier determinations of n.

Chapter 3 TYPES OF NUMBERS
With this value of n, al-Kashi calculated the error to be
expected in the circumference of a circle with a diameter
600 000 times that of the Earth - provided that this were known
with the same accuracy - and found it to be less than
"the thickness of a horse's hair" - an old Persian measure,
& 0.7 mm.
al-Kashi evidently had a good understanding of the mean-
inglessness of long chains of decimals.
European Polygonitis
The race for more n decimals did not start in Europe until
1593 when - evidently unaware of al-Kashi's result -
Valentinus Otho determined n to six, Francois Viete to nine,
and Andriaan van Roomen to sixteen decimal places.
The hunt still went on using polygons with an ever-increasing
number of sides. Best known are the results of Ludolph van
Ceulen - a fencing master teaching arithmetic, surveying,
and fortification at the engineering school at Leyden in
Holland - who in 1596 gave 20 decimals of n in Van den
Circkel, followed by 32 decimals in Arithmetische en
Geometische fondamenten, published posthumously in 1615,
and 35 decimals published in 1621 by his pupil Willebrord
Snel,
n = 3.141592 653 589 793 238 462 643 383 279 502 88 ...,
the last three digits of which were engraved as an epitaph on
van Ceulen's tombstone.
VANDEN CIJICKEL,
33aec in gfjelxcrt toetbt te bmfcm
looaveuct Don alls Caduff (met allt fiqutuB/Vta lanon rate ocotnnu
XanmbtOotm) mhtqlpmxun tammmmSin. 3cnn/tiierjfiBnmn*
• tp&m m tun CteMbrCctettun/bsffinmnftt urn cm t,4./,iv$aaJM
H3cranan4trtcta|[mtt(Qcna(ycii/al $auu & ftQmmd
fcanftnt-tnipfottfjajdan. 3tatvDti r,»Mi.*MMifcom-
tptutiauurnuCpomatttiCatiOmTnmbt^haxvmtk*
fcrt 23ogtp ctoat tntt CtaomtiQaaxmjbtamiim/
•cc Aut eirtu bc&aafcm.
Kochd«Tt/cfca Si^vtu, TtNOlNttTU, cait Sicuntivu,
act ha |hcbiuyck ?*n dica, hoofh>aoodigh voor 4c land, mami Met ral
tmdtn koaalfbc <hutea.d.€r|hciiick« «oy t ia drack uyvgliffCTta.
XmIactaavban3^atmamctamfft)andcZ^£dmDacrtoe
0un<niUing(^gttmi«k&oai>mtCMUhyCnwp«iaigri«»w
' nbtoooi tBjjrtjttUuwrh tamrfra/ cuEttgftippotr.
TwreU L&tU.
lUnnintf atmtovma»!Mm«iUn<tta*itt<flitttf<itfcigritqm/ ctfa
tr<ta.Jttta«d)»fll
AUadoor Lf DOlf TuCt* UN, {tboitaia KtlOiiMlTu,
btfcbx*vni, adc la iiack gheboche
tSfaftpsfcC *»* Ifpfim/ 3p loos Abnbamft. Banficr Manic*
Boo; 100ST ran COLSTER 23orth-DfKocptt/
Anno lit;.

Section 3.6 The Quest for n
93
Ludolphine Constant
van Ceulen's accomplishment so impressed his
contemporaries that n was often called the Ludolphine constant.
The Archimedean approach to more and more decimals of n
continued both within and outside Europe, but added nothing of
theoretical importance or practical value.
VIETE Francois
(1540-1603)
EULER Leonhard
(1707-1783)
WALLB3 John
(1616 -1703)
Exit Poly gonitis, Enter Analysis
The first to devise an infinite product for n was the French
mathematician Francois Viete with the formula
n 1
published in 1593 in Variorum de rebus mathematicis respon-
sorum liber VIII. The formula was obtained from a series of
Archimedean polygons, starting with a square.
Calculation is easier with the trigonometric version of the
formula given later by Leonhard Euler,
n sin (rc/2)
2 cos (7c/4) • cos (7c/8) • cos (tc/16) ...
The development of differential and integral calculus in the
second half of the 17th century replaced the geometric methods
of calculation by analytic methods using infinite products,
continued fractions, and infinite series expansions of inverse
trigonometric functions, permitting the calculation of n to any
degree of accuracy. The formula
k 2-2-4-4-66-88...
2 "1 - 3-3-5-5-7-7- 9 • ... '
built on integration of trigonometric functions, was presented
in 1655 by the English mathematician John Wallis in
Arithmetica infinitorum. Here, for the first time, we have an
expression for n as a product of rational numbers.
GREGORIE James
(1638-1675)
von LEIBNIZ Gottfried Wilhelm
(1646 -1722)
Plain Digit Hunting
Interest soon focused on the series expansion of the arctan
function,
arctan x = x — — + —
6 o
x{
(- 1 < X < 1),
found in 1670 by the Scottish mathematician-astronomer
James Gregorie and, independently, by Gottfried Wilhelm
von Leibniz in 1673. With x = 1, the series simplifies to
71 , 1 1 1
4=1"3+5-7+
Since this series converges only very slowly,
mathematicians looked for more rapidly converging expansions,

Chapter 3 TYPES OF NUMBERS
generally in the form of sums or differences of arctan
functions, starting another craze for ever-lengthening trains of
decimals:
In 1705 Abraham Sharp computed 72 decimal places,
1706 John Machin 100
1844 Zacharias Dase 200
1847 Thomas Clausen 248
1853 William Shanks 527
1945 D. F. Ferguson 710
1947 J. W. Wrench, J/
L. R. Smith
808
spending endless hours in filling reams of writing paper with
innumerable calculations of no practical value.
Computeritis
And there, perforce, human digit hunting came to an end
because of the appearance of an immensely more powerful
competitor - the electronic computer - with which no human
brain could hope to compete successfully in speed and in
memory capacity.
The first computer calculation of n in 1947 produced 2037
decimals in 70 hours of machine time; in 1955 the result
improved to 10 000 decimals in 100 minutes.
100 000 decimals was reached in 1961; 250 000 in 1966; 500 000
in 1967; 1 000 000 in 1974; and in February 1992, 2180 million
decimals.
In 1995 the fastest methods for computer calculation of n were
realized with the help of elliptic functions.
With today's computer technology, there is no limit to the
number of decimals of n that can be obtained except that set by
time and money.
Besides the satisfaction of setting a record, and a faint hope for
a chance appearance of some sequence of digits repeating
itself - albeit irregularly - and providing a means for testing
speed and accuracy of computers, the hunt for more decimals
of the value of n makes no practical sense.
Scriptural n
In 1 Kings, 7.23, we find the following verse describing a large
vessel in the courtyard of King Solomon's temple:
And he made a molten sea, ten cubits from the one brim to the
other; it was round all about, and his height was five cubits: and
a line of thirty cubits did compass it round about.
From this we infer a value of n = 30/10 = 3 .

Section 3.6 The Quest for n 95
In his textbook on geometry from around A.D. 150, Rabbi
Nehemiah taught:
To measure the area of a circle, multiply the diameter
into itself and throw away from it one seventh and the
half of one seventh; the rest is the area.
This is, in today's notation,
where d denotes the diameter, r the radius.
Nehemiah tells us, if we may trust the sources, that the
scriptural circumference is measured on the inside of the wall
of the vessel whereas the "people of the world" measure their
secular circumference on the outside.
The difference between scriptural n (= 3) and secular n (= 3 1/7)
is - according to the learned rabbi - explained by the
thickness of the wall which accounts for that seventh.
This explanation leaves us at a loss, however, to know whether
the volume of the vessel - 125 n - may have been a sacred 375 or
a worldly 398 cubic cubits.
7Cragmatic 7tractitioner's 7treci7iitous 7Clunge
The established fact that n is transcendental has not
discouraged self-styled thinkers from believing otherwise.
Thus, it so happened, in 1896, that a physician of Indiana in his
Solitude had given much thought to the age-old problem of
quadrature of the circle. One of his thoughts was that this task
might be simplified if one could only find a more propitious
value for n, such as
n = 3.2 .
The good doctor wanted to make his great invention available
to all schoolchildren, real estate agents, bureaucrats, and their
likes, in Indiana and in the Great Big World outside - for a
consideration, of course.
With visions in his mind of great tsunamis of dollars rolling
in, he let it be known - being a good patriot - that he intended
not to levy any fees on the use of his n in his home state,
reserving for himself only a small cut of monies accruing
from abroad.
Conventional circle Indiana circle
n = n n = 3.2

96 Chapter 2j TYPES OF NUMBERS
Moving in the very best circles in Indiana, Doc 71 managed to
square, if not actually a geometric circle, at least some exalted
personage to launch a state of Indiana bill laying down that n
be henceforth equal to 3.2, exactly.
The bill was carried unanimously by a 67 to 0 vote in the House
of Representatives, and thus was well on its way to becoming
law - to wit, a law of Indiana, not a law of nature.
By a fortunate circumstance, a Purdue University professor of
mathematics happened to be visiting the statehouse to lobby for
an appropriation for his alma mater when the Senate was about
to debate the proposed mathematic legislation.
During a lull in the debate he succeeded in convincing some of
the senators of the horrendous absurdity of the bill - they might
just as well pass a law that the Earth is flat (at least in
Indiana) - and the debate of the bill was deferred "until a later
date".
It still is.
Vivitur ingenio, csetera mortis erunt.
"Genius lives on, all else is mortal."
VESALIUS Andreas From Andreas Vesalius, De humani corporis fabrica (1543).
(1514 -1564) Flemish anatomist

97
Chapter
4
CORNERSTONES OF MATHEMATICS
***
4.0
4.1
4.2
4.3
4.4
24.4-
4.5
24.6
4.6
4.7
4.8
Beginnings
Symbols Galore
Fundamental Operations
Laws of Arithmetic and Algebra
Powers and Roots
.5 Powers and Roots of Complex Numbers
Logarithms
Logarithms of Complex Numbers
Mathematical Proof
Reliability of Digits and Calculations
Simple Calculating Devices
Page
98
101
113
132
134
791,793
150
794
157
161
168
Indicates cross-reference

98 Chapter 4 CORNERSTONES OF MATHEMATICS
4.0 Beginnings
Egyptian Papyri
Our knowledge of ancient Egyptian mathematics has come to
Rhind Papyrus us mainly from the Rhind Papyrus — now in the British
Museum - named after the Scottish archaeologist and
antiquary Alexander Henry Rhind, who purchased it in 1858
at Luxor in Egypt.
The scroll is about half a meter wide by nearly five and a half
meters long with many important fragments missing. By a
lucky chance, however, several fragments were found in 1922
in the archives of the New York Historical Society, which had
acquired them along with a noted medical papyrus.
The papyrus contains mathematical tables for the calculation
of areas and the conversion of fractions, elementary
sequences, linear equations, geometric problems, and extensive
information about measurements.
The "title page" contains a preface which dates the copy of the
scroll to "the fourth month of the inundation season of year 33,
when 'A-user-Re' was King of Egypt". This means that the
scroll dates from about 1650 B.C., when Ahmes the Scribe copied
it from an older document dating back to the Twelfth Dynasty,
1849 - 1801 B.C.. In honor of Ahmes the Scribe, the scroll is
sometimes referred to as the Ahmes Papyrus.
Detail from the Rhind Papyrus. © British Museum.

Section 4.0 Beginnings
99
1 tl cc/
1 2 3
:u 4> X
6 7 8
Plus Symbol
Minus Symbol
Equality Symbol
un *i
4 5
€» A
9 10
T. Eric Peet, The Rhind
Mathematical Papyrus, 1923
cf. p. 34
The Rhind Papyrus is in hieratic script, a form of writing
developed from the pictographic hieroglyphics found on
monuments and other Egyptian artifacts. The papyrus contains the
earliest known symbols of mathematical operations:
- plus is denoted by a pair of legs /X walking toward the
number to be added, with the general meaning of "to go
up";
- minus is a pair of legs walking away from the number
to be subtracted, meaning "to go out" in everyday language;
- the result of an operation is identified by the sign r ^ j ,
meaning "together" - the earliest known equality sign, but
placed at the very end of the problem;
- an unknown quantity is always indicated by the character
^fi, meaning "heap".
Translated from the hieratic script into pictographic
hieroglyphics, the section of the papyrus marked by an X at the right-
hand edge becomes:
JLl
9(lA
•A
in
a-JLY
Moscow Papyrus
The problem reads, from right to left:
"Two-thirds of an unknown added [to the unknown], threa-
fourths of the unknown taken away [from the unknown],
10 remains",
which means, in modern notation,
2 3
Another notable Egyptian papyrus containing mathematical
problems is the Moscow Papyrus - named after its current
location. It dates from around 1850 B.C., that is, about the same
time as the document copied by Ahmes.
The mathematical symbols used in the Moscow Papyrus are
the same as in the Rhind Papyrus.
Scribes at work. From a relief on an Egyptian grave.

100
Chapter 4 CORNERSTONES OF MATHEMATICS
Arithmetic, Logistic, Algebra
Greek, arithmetike, from
arithmos, "number", and
techne, "science"
Greek, logos, "reckoning",
"account", "reason'
al-KHOWARIZMI
(9th century)
In ancient Greece, arithmetic meant theoretical work
involving numbers, while the practical everyday calculations of the
merchant were called logistic - a distinction that persisted
well into the early 16th century. From then onward the term
arithmetic has generally been used for both practical and
theoretical operations involving numbers. Logistics, in the
plural, has become a predominantly military term.
The advent of Islam, and the Arab conquests during the 8th
century, led to Arab acquisition of Greek and Hindu
scientific writings, many of which would have perished but for their
translation into Arabic.
Most important of the mathematical writings of this era were
the works of the Persian al-Khowarizmi, who around 825, in
Baghdad, wrote the
Hisab al-jabr w'al-musqabalah y
meaning literally,
"Science of the Reunion and the Opposition",
or more freely rendered,
"Science of the Transposition and Cancellation",
a monograph which became most influential in the
introduction of algebra in Europe by the Moors in Spain.
The title refers to the two principal operations used by
al-Khowarizmi in solving equations:
al-jabr
al-musqabalah =
the transposition of terms from one
side of an equation to the other;
and
the cancellation of equal terms
appearing on opposite sides of the
equation.
The relation
bx + q = ax2 + bx - 3q
transforms by al-jabr to
bx + q + 3q = ax2 + bx
and by al-musqabalah to
4q = ax2 .
Arabic, jabara, "to set",
"consolidate"
The Arabic al-jabr was latinized to algebra.

101
There, once was a fellow in Mosj0en,
Who felt math was lost in the ocean.
He wrote symbols galore
- A thousand pages or more -
So folios would acquire the notion.
4.1 Symbols Galore
We can distinguish three periods in the development of
mathematics:
o the rhetorical phase when all words relating to
mathematical operations and descriptions were written out in
full;
o the syncopation phase when abbreviations of these words
were beginning to be used; and
o the symbolic phase when abbreviations were fading out
and giving place to symbols.
Mathematics symbols are of three kinds:
o Symbols for numbers, quantities, variables, or objects.
Examples are symbols used for trigonometric functions,
powers, roots and logarithms and their indices and
exponents, and symbols used for variables.
o Symbols of operation, which describe things to be
performed. These symbols include symbols indicating
addition, subtraction, multiplication, division, and their
attendant grouping symbols, summation and product
symbols and factorial notation; also derivation and
integration signs and certain other function symbols.
o Symbols of relation, which describe things established:
symbols of equality and inequality, of ratio and proportion.

102 Chapter 4 CORNERSTONES OF MATHEMATICS
We tend today to regard the meaning of established
mathematical symbols as non-negotiable and inviolable, but this
has not always been so.
In times past,
o an x might have stood for addition;
o a "plaine lyne" — could have meant division;
o a double line = might have meant subtraction;
o multiplication and division could both be denoted by a dot,
on the line or raised halfway above it, not forgetting that
the same dot might also have been a decimal point;
o and the signs °° and <* both meant equality, when today <*
signifies proportionality and °° is the designation used
for infinity.
It must have led to many interesting debates when every
worker in the mathematical vineyard could make up his own
arsenal of symbols:
- "plus" might have been p or P or p or e or —|— or 4- or _|_
or^« , or even +;
- "minus" might have been m or m or 3e or — or — or -=- or
-r or $— or I — | or -h- ;
- "equal to" might have been oo or ex or ^> or [or || or =
or £ , or even =.
Symbols of Addition and Subtraction
For addition, the ancient Greeks simply juxtaposed the terms
to be added together, without a connecting sign - as we are still
doing today: 4^=4+^- For subtraction, Diophantos of
Alexandria seems to have used the symbol a, believed to be an
inverted and truncated letter \j/ (psi).
Early European - Italian - symbols for plus (or piu) were P, p,
and p, the last of which was the more common; symbols
for minus (or me no) were M, m, and m, the bar as usual
indicating omission of one or more letters.
PACIOLI Luca Luca Pacioli wrote p and sometimes e for plus, and m or de
(1445-1517) (Latin demptus, "taken away") for minus; this usage is found
in many works in the 15th and 16th centuries.
The word plus was also used in full, but not before the end of
the 15th century, considerably later than its fellow minus.
which appeared in 1202, in Fibonacci's Liber abaci. German

Section 4.1 Symbols Galore
103
REGIOMONTANUS Johannes
(1436-1476)
and Dutch writers continued to be rhetorical and wrote signum
additorum and signum subtractorum.
The word et, Latin for "and", was found in many manuscripts,
generally in a contracted form closely resembling the symbol
+, leaving little room for doubt that + is a ligature for et.
The origin of the minus sign is less clear, but it may well be
that it is the remainder of an m whose bar has "moved down"
and become a symbol of its own.
As far as we know, the symbols + and - first appeared in
1456 in an unpublished manuscript by the German
mathematician and astronomer Regiomontanus (Johannes Miiller,
or Molitoris). Their first appearance in print is in the
mercantile handbook Behennde unnd hilpsche Rechnug auff
alien Kauffmannschaften ("Neat and Handy Calculations for
All Tradesmen") by Johannes Widmann, printed in Leipzig
in 1489.
THE GROVND OF
JtorMnf (be ttootae «nb practifc of
flrit&tnrnhe, aotb in tbbolstrambirs
MO Jeactiont, iftrr a mote rafper
ano eracttr four, torn any* IjHe
bact) tjf toctto Dcrnc
ftttoutr.ftutiDu
oara uctto CO*
otttons*-
ffUocbpai.A. obuti
UCOKOB
Dotto) of pbpfiUt.
1558 edition
RECORDE Robert
(c. 1510-1558)
$&ti)tt\ttot Diwo
fyamee
p ■
44
7X
±
±
+
4 -f- J Tffilttl^
4 T^ i* ttTfftftad
&$ofwne
mirtwif
44 ixriri. ujcr
*9 J154JJ4
jemflte* ftaft wino 4as4~ tatt i)T mcr
griZ4ti vtifcjtft 151110( X4*vrim%fet
jtsifi oar ;u arorrty —ajifirf iff
mno m«rwn j S 7 5>c fubtratNr vonit
4 f * 9 'GJtmOpIrFbM i f z IB 11a
The above text concerns thirteen chests of goods assigned a certain
price per hundredweight (cwt). The weight of each chest is given in
whole cwt plus or minus a number of pounds.
The chest at the top weighs 4 cwt plus 5 lbs, the next chest 4 cwt less 17
lbs, etc.
Thus, the symbols + and - have no operational function, but merely
indicate excess or deficit.
In England, Robert Recorde, influential author of
mathematical textbooks, in 1543 published a textbook in arithmetic,
Ground ofArtes, in which he states that
thys figure, +, whicfie betokgneth to muche, as this tyne, -,
ptaine without a crosse tyne, betofgneth to tyttte.
As symbols of operation, most English writers reserved the
notations + and - for use in algebra. They did not acquire
today's operational functions until around 1630.

Chapter 4 CORNERSTONES OF MATHEMATICS
± + These pairs of symbols have several uses.
The notation a ± b implies that two cases exist: a + b and a - b,
with different numerical values except when b is 0.
The notation ± is also used to indicate reliability of the digits
in a number and limits of inaccuracy of measurements.
A further use of the plus-minus and minus-plus notations is to
indicate correspondence between mathematical expressions;
instead of writing the two similar statements
cos (G + 0) = cos G cos 0- sin 6 sin 0
cos (G- 0) = cos Gcos 0 + sin G sin 0
we may combine them to one,
cos (G± 0) = cos 0cos 0+ sin G sin 0
where upper signs are taken together, and lower signs together.
A The increment A x, "delta x", suggests a small amount added
to, or subtracted from, a given value of a variable x ("change
in x"). The increment is usually understood to be a minute
quantity in comparison with the given value.
In mathematics, an upright typeface is used for the increment
notation. The notation A (sloping delta) is generally used in
physics to imply quantity excess, or difference in quantity
(e.g., mass excess, Am, and temperature difference, AT).
Symbols of Multiplication and Division
Symbols of multiplication and division were rather late in
developing compared to those of addition and subtraction.
The symbol x for multiplication was introduced by the English
mathematician William Oughtred in 1631 in his Clavis
mathematicae ("Key to Mathematics"). As a symbol, it was
not exactly new having been employed in so-called "crosse
multiplication" when dividing fractions.
It was not readily accepted by arithmeticians, however, and
did not come into general use in textbooks in elementary
arithmetic until the latter half of the 19th century. Nor was
it well received by algebraists because of its resemblance to
the variable x, and so the dot - suggested by the English
mathematician and astronomer Thomas Harriot in his Artis
analyticae praxis, published posthumously in 1631 - came
to be employed. Adriaan Vlacq, the Dutch publisher and
computer of tables of logarithms, had suggested a dot in his
Aritmetica logarithmica in 1628, though not as an active
symbol of operation.
The first mathematician of undisputed prominence to use the
dot in a general fashion for algebraic multiplication was
Leibniz in 1686; later algebraists used the dot in cases where
the absence of a symbol would not have sufficed.
OUGHTRED William
(1575-1660)
HARRIOT Thomas
(c. 1560 -1621)
VLACQ Adriaan
(1600 -1666/67)
von LEIBNIZ Gottfried Wilhelm
-1716)

Section 4.1 Symbols Galore
105
RAHN Johann Heinrich
(1622-1676)
The earliest method of writing a division or a quotient of two
numbers was by placing the dividend above the divisor, thus
312
256
without any special division symbol between them, as in the
Bamberger Rechenbuch of 1483. A fraction bar was later
introduced between the numbers,
312
256'
and finally the numbers themselves disappeared, leaving the
symbol -=■, where the two dots remained to represent the
numbers.
The symbol -7 appeared in print for the first time in Teutsche
Algebra (Zurich, 1659) by Johann Heinrich Rahn; its first
mention in English is in the 1668 edition of the translation of
Rahn's algebra.
The symbol -r had long been used in continental Europe and in
Scandinavia to indicate subtraction, a use now fast receding.
In English-speaking countries it always denotes division,
and is generally found on the division key of electronic
calculators.
Summation and Product Symbols
EULER Leonhard
(1707-1783)
DESCARTES Rene
(1596-1650)
Greek capital letter sigma, Z, denotes summation.
If we let a/ (read "a subscript i" or "a sub i") be an algebraic
expression, the notation
n
I
1 = m
<*i ,
which reads "the sum of all a ; as i goes from m to ra", is a kind
of shorthand notation for the sum
<*m + <*m+l + am + 2 + ... + an_\ + an.
11 Greek capital letter pi, fl, denotes a product,
n
n
1 = m
a;
1 )
"the product of all a i as i goes from m to ra",
am ' am + 1 * am + 2 • • • • * an - 1 • an •
The summation notation was introduced by Leonhard Euler,
the product notation by Rene Descartes.
The letter i is called the index of the summation, or index of the
product; the letters m and n define the limits of the index, m
being the lower limit and n the upper limit of i.

1 06 Chapter 4 CORNERSTONES OF MATHEMATICS
The use of the letter i is in no way crucial; j9 k, and I are also
commonly used as well as the Greek letters i, v, and \i. The
index should not be confused with the imaginary unit, which is
preferably written with an upright i.
5
^(3i + 4) = (3x2 + 4) + (3x3+4) + (3x4 + 4) + (3x5 + 4)
j = 2
= 10+13 + 16 + 19 = 58
5
TT(3i + 4) = (3x2 + 4)(3x3+4)(3x4 + 4)(3x5 + 4)
i = 2
= 10 x 13 x 16 x 19 = 39520
When there is no risk of ambiguity or misunderstanding, the
notations may be simplified; thus
n n n n
xai => Xa* and nat ^ Y[ai
i=0 0 i=0 o
Factorial
The factorial notation n ! represents the product of all positive
integers from 1 to n, inclusive,
n\ = 1.2.3.4. ... .(ti-1).7i,
and thus may be considered a special case of the II product
notation.
We have, by definition,
(n !) (ti + 1) = (n + 1)!
known as the recursion formula; inserting n = 0, we find that
we should define factorial zero as follows:
6! =
5! =
4! =
3! =
2! =
1! =
0! =
0! = 1.
6x5x4x3x2x1
5x4x3x2x1
4x3x2x1
3x2x1
2x1
1
1
= 720
= 120
= 24
= 6
= 2
= 1
= 1
As we go down the lines, we find the next one by dividing, in
turn, by 6, 5, 4, 3, 2, and 1.
The factorial symbol was introduced in 1808 by Christian
Kramp in order to avoid printing difficulties caused by the
earlier symbols \n and rj\ .
n
KRAMP Christian
(1760-1826)

Section 4.1 Symbols Galore
107
Symbols of Equality
RECORDE Robert
(c. 1510-1558)
W)t tulictftonf
oftottte,
io&i'c &e is tit reconDe parte of
ArithmetiJccrconttltapng tljmrac*
tiou of ttootet: &bt OtfiHg p;amfe,
loitfc tbe nils of if *u/;«»:anD
t^tDOo;fcrsof5ar^
The symbols following the
number symbols indicate
variables and constants,
and correspond to
14jc+15 = 71; x = 4
20x-18 = 102; x = 6
In manuscripts dating from before the advent of printing, and
the first hundred years afterwards - until about 1600 - the
equality of two mathematical expressions was denoted by
the word aequalis or one of its inflected forms - written out in
full.
From then on, until around 1680, various abbreviations of
these words were used, or symbols chosen to signify equality,
and generally distortions of the initial letter se of the word
aequalis. Descartes, as late as 1637, used °° to denote equality.
Robert Recorde, in his textbook of algebra, The Whetstone of
Witte from 1557, introduced two long parallel lines as a
symbol for equality.
The Jrte
as tljcir tuojfcf0 toe crtcntsc) to Dittincf c ft oncly fttf o
ttuoopartcf. Qfllbcrcoftbcfhttcu, -vrbenonenomberii
tqutlle vnto one etber. 3nfe tf)C ftcoitfec (s >T>bcn one noms
beris compared as equalle Vnto.%otbcr nombers.
flltuatca tuUlpng pou to rcmebcr, tbat pou refeuce
£ournomber0 , to tbcir Italic fccnominationu, aim
fmallctte fo;me0,licfo?c pou pioccfcc anp fartbcr.
anfc agam,if pour edition be foebe, tljat tbc grca>
tcttc Denomination G/?/(r, be f oincfc to anp parte of 4
compound uomder, pou l^all tourne it fo, tbattbe
nombcroftbcgrcatcttc Ogncalone, maicttaitfccas
cquallctotbcrcttc.
anD tbu$ 10 all tbatncatetb to be taugljtc, couccr/
npngtbiaiuoojUc.
^olubc(t,fo; caffc altcratf 0 of rf Mf/wi.) Ml p;o>
pounce a fetuc crapte0,lrtcaufc tbc extraction oft^ctc
rootc09maic tbc mo;c aptlp Dec tujougbte. aitfe to a>
uoidc thctcutoufc repetition of tbcfc tuoo;fcc0:i0c*
quallc to: 3 tutU fettc a* 3 doc often in tuoo;fec tifca
patrc of parallclc0,o; <25cmotue lines of one lengtbe,
tl)us:=* ,lrieaufc noc.2. tbpiigc0,can lie moars
cqualle. antmotumarfcetbcfeitomljcr*.
1.
2«
14.^.-4-. 1 j.^^w?1#£
20¾. #is.^=-r=^i o2.f.
Recorde provides an excellent argument in favor of his choice
of symbol:
... And to auoide the tediouse repetition of these zvoordes:
is equalle to : I zvillsette as I doe often in zuoorfg. use, a
pair of parallels, or Qemozue lines of one lengthe, thus:
============== bicaxise noe .2. thynges, can be moare equalle...
Gemowe is a distortion of Old French gemeus, meaning twin.

108
Chapter 4 CORNERSTONES OF MATHEMATICS
BUTEO Johannes
(c. 1492 - c. 1568)
XYLANDER Giulielmo
(1532-1576)
def
Recorde's proposal for an equality symbol was not without
competition, however. In 1559, Johannes Buteo, otherwise
Jean Borrel, in his Opera geometrica suggested a bracket [;
and in 1575 Xylander, or Wilhelm Holzmann, in Arithmetica
suggested two parallel vertical lines
This modern equality notation means that two mathematical
expressions are equal.
It it also correct to employ this notation in the case of decimal
fractions ending in three dots, indicating that further digits
would follow, e.g.,
71 = 3.141592 653... or 3.14....
If the absence of further decimals has not been indicated by the
three dots, the notation for approximate equality should be
employed, e.g.,
n * 3.14 .
An identity is a statement of equality which is true for all
values of the variable, e.g.,
(x + y)2 = x2 + 2 x y + y2 .
This identity notation may be replaced by the equality symbol
=, and the relationship is still valid.
This rather unusual symbol establishes the fact that the
equality is by definition, e.g.,
a0 dJLf 1; 0! d|f 1.
0! d_tf 1 has a different meaning than 0 ! = 1; the first must be
taken as true, the second might be false.
According to the fundamental laws of arithmetic and algebra,
the same number or quantity may be added to, or subtracted
from, both members of an equation without affecting the
equality. Nor will multiplication or division of both sides of
an equation with the same number influence the correctness of
the equation.
Symbols of Inequality
In mathematics, symbols are sometimes required to describe
the relative magnitude of two or more quantities, even as a
"degree of inequality".
^ This symbol signifies inequality and is thus absolute in its
meaning of "not equal to". It is often employed to emphasize a
condition necessary for a certain statement to apply,
"a I b is defined only if b * 0".

Section 4.1 Symbols Galore
109
>
HARRIOT Thomas
(c. 1560 -1621)
OUGHTRED William
(1575 -1660)
These symbols mean greater than and less than, and were
introduced by Thomas Harriot in his Artis analyticae praxis,
published posthumously in 1631.
Harriot's symbols were not used very frequently until the
early 1700s; many mathematicians and printers preferred the
symbols L-.. "" and ; suggested by William Oughtred
in 1631 in his Clavis mathematicae. The symbols reappeared
slightly modified as U-, for non majus and f for
non minus in the 1647 edition of Clavis, with the meanings not
greater than and not less than, respectively; these definitions
remained in following editions.
Guilelmi Oughtred
AETONENSIS
quondam Coliegii Regalis
in C amtae a.i o i a Sod!,
CLAVIS MATHEMATICS
DENVO LIMAtA.
Sire pedw
FABRICATA.
Cum aliis quibufdam quflem
Commenucionibus, quae in fe«
quenci pagina rcccnlcncur.
. Edkio tertia au&ior Sc emenduior."
Third edition, 1652
OXONU
ExcudcfaacLioN.Licirtai.0, Vaminc
•pudTao. Rot ttitoK. 165 s.
9
,ffquale=.
Majus C".
Minui -3*
Nonmajui
Non minus
Simile Sim.
Proxime majus c-.
Proxime minus-a.
£qualevel minus c,
£quale vel majus "^3
Proporcio, five ratio zqualis ::
Major ratio vr • Miner ratio -
Continue proportiomles "•i*.
Commenlurabilia "°—
Incommenflirabilia "Q-.
Commcnfurabilia potemia ^-.
fncommenfurabilia potcntia ^-.
Rationale, £**»*, R, vei it.
Irrational e, *A.«>*r, V\
Medium five mediale /iT
Linea fefta fecundum excreraam 7
6c mediam rationed j
Major ejus portio t
Minor ejus portio t.
2<ftA + E. gefta+e.
X eft A-L £ eft a*
> <
BOUGUER Pierre
(1698-1758)
* f
» «
Unconditional Inequality
Conditional Inequality
These symbols mean greater than or equal to and less than or
equal to; they are fairly late additions to the family of symbols
and were written ^ and ^ when introduced by the French
scientist Pierre Bouguer in 1734.
These rather unusual symbols mean not greater than and not
less than, respectively. We note that j> and < have the same
meaning.
These notations, much greater than and much less than, may
be used to compare inequalities, e.g., a » b ; c « b.
Like equations, an inequality is unaffected if the same
number is added to or subtracted from both sides or if each side
is multiplied or divided by the same positive number.
However, if each side of an inequality is multiplied or divided
by a negative number, then the sense of inequality is reversed.
For example, if (x + 2)2 > 3, we have
4[(x + 2)2 -a] > 4(3-a); -4[(x + 2)2-a] < -4(3-a).
An unconditional inequality, e.g., 5x2> 0, is true for all
values of x, whereas a conditional inequality, e.g., 5x > 0, is
not.

110
Chapter 4 CORNERSTONES OF MATHEMATICS
Symbols of Ratio and Proportion
Antecedent
Consequent
Nichomachos, in ancient Greece, looked upon ratio as a
feature of arithmetic, Eudoxus as one of geometry; medieval
writers regarded ratio and proportion as quite distinct from
both arithmetic and geometry. Today, ratio and proportion are
considered primarily part of algebra.
The actual meaning of the word ratio - from the past participle
ratus of a Latin verb meaning "to estimate" - is reckoning,
calculation, relation.
Ratio
The ratio of two numbers, or quantities,
t- , a I b , or a : b ,
which reads "the ratio of a to 6", is a measure of how
many times a contains b. Thus, the terms ratio, quotient, and
fraction are synonymous with the result of a division, a
difference being that a ratio is generally not expressed in the
form of a decimal fraction.
The first term of a ratio is called the antecedent; the second
term is the consequent. The antecedent equals the ratio times
the consequent.
This notation is used exclusively to express a change of scale.
In cartography, a scale of 1 : 100 000 means that 1 cm on the
map corresponds to an actual distance of 100 000 cm, that is, 1
km. This may be written
1 cm = 1 km ,
which reads "1 cm corresponds to 1 km". Another example is
212°F = 100° C.
OUGHTRED William
(1575-1660)
Proportion
When two ratios alb and eld are equal, the four terms a, b, c,
and d are said to form a proportion, or to be in proportion.
Originally written
a-b-c—d,
this was changed in 1631 by William Oughtred in his work
Clavis mathematicae to
a . b :: c . d
and in 1657 in Canones sinuum, a trigonometrical text, to
a : b :: c : d ,
a form of notation which was used well into the 20th century.
As there is no sense in having a different notation to signify
equality in a proportion, today we write
L ^ a C
a: o = c : a . or— = -r ,
b d
which can be read "a is to b as c is to d".

Section 4.1 Symbols Galore 111
The terms in a proportion are called proportionals; a and d are
known as the extremes, the intermediate ones b and c are the
means.
We find by cross multiplication that ad = be; thus:
o The product of the extremes equals the product of the means.
This rule has been known under many different names:
Regula de Tri, Regula de Tribus, Rule of Three, Golden Rule,
Merchants' Rule, Merchants' Key, etc. It was known by the
Hindus in the 6th century A.D. and seems to have been of
mercantile concern till the end of the 15 th century, surrounded
by mystique.
If the two means are equal,
a m
m d '
m is the mean proportional, or geometric mean, of a and d,
m = yad ; thus:
o The mean proportional is the square root of the product of the
extremes.
Two variables x and y may be so related that the ratio of one
to the other is always the same, x/y = constant, which may be
written
x oc y, or x ~ y
"x is proportional to y".
In geometry, ~ has the meaning of similarity, that is, "not
differing in shape but only in size".
Intervals
If a variable x satisfies both inequalities a < x and x < b, where
a and b are constants, then
a < x < b
and x is said to be located in the closed interval a and b; this
interval [a, b] includes a and 6, if a < b and is empty otherwise.
If, on the other hand,
a < x < b ,
x is located in the open interval a and b\ this interval ]a, b[
extends up to both a and b but includes neither.
If x satisfies either
a < x < b or a < x < b,
it is located in a half-open interval [a, b[ or ]a, b].

112
Chapter 4 CORNERSTONES OF MATHEMATICS
On the number line, open ends of intervals are marked by
open circlets, closed ends by filled-in circlets.
h,bl
}a,b[
a
a
la.b[
—4=
+
]a,6]
a
a
The use of parentheses instead of reversed brackets to indicate
open ends of intervals is also common; thus,
(a, b] = ] a, b] ; [a, b) = [a, b[; (a, b) = ] a, b[.
Closed ends split a and 6:
JQ
£Z
££
JQ
If both ends open, a and 6
can split a pot of tea:

113
4.2 Fundamental Operations
Most textbooks recognize four fundamental operations of
arithmetic: addition, subtraction, multiplication, and
division, traditionally known as the four simple rules of
arithmetic. Of these, subtraction can be considered the inverse of
addition; multiplication by an integer is a repeated addition
of like terms, and division the inverse of multiplication and
related to repeated subtraction.
To these four operations we add three, making seven
fundamental operations in all: raising to integer powers, which is a
repeated multiplication of a given number of like factors; and
its two inverse operations - root extraction, and the calculation
of logarithms.
RIESE (RISEN) Adam
(1492 -1559)
<£$tnbu$/auff2ini€n
i^imm i dtfcgAfftoi win* fouffrtan*
tmD smcfct iumacfyn/ au$ Don <Ouat>raf /
fm/Hfat*uu. * *""
fxnwt&£om$irf.
Title page of the 1574 edition
of Adam Riese's Rechenbuch,
first published around 1520.
Riese's books were of great
importance in spreading the
knowledge of arithmetic
throughout Europe; "according
to Adam Riese" became a
household word guaranteeing
accuracy of mathematical
operations.
Sr4iicF.25ty.<r&r.<S3ffl.<£ir&M. i y 7 4<

114
Chapter 4 CORNERSTONES OF MATHEMATICS
Constants
Unknowns
Coefficients
Variables
Algebra- the latinized form of Arabic al-jabr - is a
generalized arithmetic which operates with letter symbols as
well as numbers: letters from the beginning of the alphabet
- a, b, c ... - are generally used to denote fixed or arbitrary
numerical quantities, generally known as constants; letters
at the end of the alphabet -x,y,z- denote unknown quantities
to be determined by solving equations.
In equations - that is, statements of mathematical equality -
constants defining multiples of unknowns, or powers of
unknowns, are called coefficients:
ax2 + b x + c = 0
T T T
coefficients constant
In equations and systems of equations containing several
unknown quantities, these are often called variables.
LA PRIMA PARTE DEL
GENERAL TRATTATO DI NV»
MfiRl.ET MISVREDI NICOLO TARTACUA,
NELLAQ.VALE IN D1ECISETTE
LIBRl SI DICHIARA TVTTl GU ATTl OPERATlVf,
PKATlCHE, XT R.BGOLE NECESJAIWB HON SOLA'
taam in tunal'im nr*oturu,& entrant fli,m» tnchor in 6gni aitn
vtt^QmtiijOua dtfciplini, dove mttrucnghi I alaito.
MALlCMiTA*
COK Li SVOl PF.IVILEGII.
Ik VititgU per Curiio Troiano dc I Naul
H D LVI.
TARTAGLIA Niccolo
(c. 1500 -1557)
The Italian mathematician Niccolo Tartaglia's General trattato di
numeri et misure ("General Handling of Numbers and Measures") was
published 1556 - 1560. Volumes 1 and 2 are considered two of the most
important 16th-century texts on arithmetic; Volumes 3-5 deal with
geometry; Volume 6 deals with algebra.

Section 4.2 Fundamental Operations
115
- YOU THINK THIS 19 EASY...
JUST WAIT UNTIL. I HAVE.
KXPLAINED IT TO VOU
Addition
Addition was sometimes called "aggregation" in the 13th
century; Fibonacci used the names "composition" and
"collection" besides "addition". The Margarita phylosophica
(1496) by Gregor Reisch illustrates addition
thus,
4679 numerus cui debet fieri additio
3232 numerus addendus
7911 numerus adductus (or collectus)
which is today:
addend
+ addend
sum
Gregor Reisch: Margarita phylosophica (1496; 1503 reprint)

116 Chapter 4 CORNERSTONES OF MATHEMATICS
The technique of an addition has not changed materially
since the introduction of Hindu-Arabic numerals.
The term "carry a digit" dates from the time of the abacus,
when a counter was moved ("carried") from the lower part
of the abacus to the space above; it appeared in English in the
17th century (Hodder, 1664); Robert Recorde writes "keepe in
mynde" to describe a mentally carried digit.
The addition of positive numbers, 2 + 6 = 8, reading "two plus
six equals (or makes) eight", and the addition of a negative
number, 2 + (- 6) = - 4, "two plus negative six equals negative
four", are illustrated on the number line thus,
2+6=8 o
2 . 6
^ -6 ° *l 2 + (-6) = -4
H 1 1 1 1 1 1 1 1 1 1 1 1 h
-5-4-3-2-101234 5678
Subtraction
Subtraction has been called by many names: deduction,
extraction, detraction, subduction; to subtract has been deduct,
extract, detract, subduce, subtray, rebate. The Margarita
phylosophica has
9 001386 numerus a quo fieri subtractio
7 532436 numerus subtrahendus
1468950 numerus relictus
GEMMA FRISIUS Reiner Gemma Frisius wrote in 1540
(1505 -1555)
numerus ex quo subductus
numerus subducendus
numerus residius
which later became and today
numerus minuendus (or superior) minuend
numerus subtrahendus (or inferior) - subtrahend
differentia = difference
Instead of difference, the terms "rest" and "remainder" may
be used.

Section 4.2 Fundamental Operations
117
Subtracting a positive number, the difference will be less than
the minuend, as shown here on the number line:
13
^} 2-6 = -4
H \-
6-2 = 4
H h
-5-4-3-2-1 0
8
The subtraction of a negative number will, instead, cause the
resultant difference to be greater than the original minuend:
6 - (-2) = 6 + 2 = 8
r*——°
■\—h
-6
-5-4-3-2-1 0
-2
■*- -2-(-6) = -2 + 6 = 4
-\ h
+
+
+
8
TIGER
On thz Mysteries of Subtraction
l.
- If you've got ten birds sitting on a tele-phone wire, and one
tat^es flight - how many will there be left?
- 9{pne. All the others will also leave.
This may not be mathematics, but is perfect logic.

118 Chapter 4 CORNERSTONES OF MATHEMATICS
Multiplication
Multiplication was thus defined by Robert Recorde (c. 1542):
(Lj = ii = 2) Multiplication is such an operacw that by Lj sumes
producyth the thyrde, zuhiclie thyrde sume so manye
times shall cotaim thefyrst, as there, are vnites in the
second.
The general form of an operation of multiplication
was, in Latin, and is, in our terminology,
numerus multiplicandus multiplicand
numerus multiplicans x multiplier
Egyptian
multiplication
1
+ 2
4
+ 8
+ 16
+ 32
93
186 +
372
744 +
1488 +
2976 +
numerus productus product
where the word product means "something produced", and has
been employed also in other operations, for instance, in
addition with the meaning of "sum". The terms multiplicand
and multiplier are today obsolescent, and are generally called
just "factors".
Most people will recall from their schooldays a nuisance
called the multiplication table. To refresh the reader's
memory, let us reproduce "the foundation of all multiplying" from
the Bamberger Rechenbuch of 1483:
©ergtunoalke 4fcultiulicitf»
1
X
3
4
%
6
A
S
9
X
*
6
8
Jo
it
U
%6
1$
*
6
9
\X
«f
13
IX
X*
XI
* $
8 io
IX l<l
%6 Xo
XO XS
%Z )o
*s K
)X *o
)* *S
6
i*
18
*4
*°
)*
zx
*8
S*
A
t*
XI
XS
V
4*
4?
$*
«
8 9
IS 18
*4 XA
$X *6
4° *S
*8 f*
f* *t
6* IX
IX 81
from which we learn, among other things, that 7 x 3 = 12 (sic!).
Printer's errors have a long history.
The earliest form of multiplication known is the Egyptian
method of duplation, which reduces multiplication to a form of
continued addition. The method - illustrated here by the
operation 58 x 93 - was frequently copied by other peoples, and
is commonly found in textbooks from the Renaissance.
58 5394

Section 4.2 Fundamental Operations 119
"Russian peasant"
multiplication
58
° 29
14
O rj
° 3
° 1
93
186 +
372
744 +
1488 +
2976 +
o
o
o
o
o
93
46
23
11
5
2
1
58 +
116
232 +
464 +
928 +
1856
3712 +
5394 5394
The "Russian peasant" method is another form of
multiplication by successive duplation and mediation. The operation
58 x 93 proceeds by successive doubling of 93 and halving 58, or
doubling 58 and halving 93, each time ignoring fractions, and
finally adding those numbers in the right-hand column which
stand opposite odd numbers in the left-hand column:
186 + 744 + 1488 + 2976 = 5394
58 + 232 + 464 + 928 + 3712 = 5394
To remove the air of mystery, let us see what actually happens.
93 = 2° x 93
= 2° + 21 x 46
= 2° + 22 x 23
= 2° + 22 + 23 x 11
= 2° + 22 + 23 + 24 x 5
= 2° + 22 + 23 + 24 + 25 x 2
= 2° + 22 + 23 + 24 + 26 x 1
2° x
22 x
23 x
2*x
26 x
58 = 58
58 = 232
58 = 464
58 = 928
58 = 3712
5394
Multiplication in its simplest form,
"four times six" 4x6 = 6 + 6 + 6 + 6 = 24
"six times four" 6x4 = 4 + 4 + 4 + 4 + 4 + 4 = 24,
is equivalent to a repeated addition of a specific number of like
terms; on the number line, it looks like this:
44,4^4,4,4^
—*. > >■ >■ ► >►
6 6 6 6 =24
-> *> »» ►
-25 -20 -15 -10 -5 0 5 10 15 20 25

120 Chapter 4 CORNERSTONES OF MATHEMATICS
If either of the factors involved is negative,
4 x (-6) = (-6) + (-6) + (-6) + (-6) = -24
6 x (-4) = (-4) + (-4) + (-4) + (-4) + (-4) + (-4) = -24,
the representation on the number line will be
.-4,-4.-4,-4,-4,-4
-24= _e -6 -6 -6
< < <-
-25 -20 -15 -10 -5 0 5 10 15 20 25
There is a choice of signs for multiplicative notation: a cross
(x), or a dot (•) placed halfway above the line. When a dot is
used as the decimal marker, the International System (SI)
recommends that it not be used to double as a multiplication
sign; the accepted notation will then be a cross,
4 x 2.5 = 10.
In algebra, when multiplying symbols - a, 6, c - or factorial
expressions, no special multiplication sign is necessary:
a x b = ab
(a + b) x (a + b) - (a + b) (a + b).
Division
Division finds how many times a given number, the dividend,
contains another given number, the divisor:
dividend . remainder
= quotient + ———:
divisor divisor
Mathematically, this is done by deducting, repeatedly, the
divisor, or multiples of it, from the dividend. If the remainder
is 0, the division is even.
Early mathematicians wrote
numerus dividendus
numerus divisor
but had no specific name for what is today called the quotient,
except descriptions such as Humerus ex divisione proueniens
or numerus querendus; the remainder, if any, was Humerus
residuus, or just residuus.
Considered a difficult operation, we meet in 1600 the following
characterization:
<<(Diuision is esteemed one of the busiest operations of
Arithmetic^ and such as requireth a mynde not
wandering, or setled uppon other matters."

Section 4.2 Fundamental Operations 121
On the number line, a division will look like this,
24/6 = 4 <6«6«6«6o
(4) (3) (2) (1)
-H 1 1 1 I 1 1 1 1 1 1 ►
-25 -20 -15 -10 -5 0 5 10 15 20 25
which shows that "24 divided by 6 equals 4".
The International System (SI) recommends two ways of
writing a division,
alb or t,
6
which read "a divided by b". Other methods are a : b and a + b;
the latter form is usually found on the division key of
electronic calculators.
Rules of Divisibility
An integer N is evenly divisible
- by 2 if it is even - this depends only on the last digit;
- by 3 if the sum of its digits is divisible by 3;
- by 4 if the number formed by the two last digits is
divisible by 4;
- by 5 if it ends in 5 or 0;
- by 6 if it is divisible by 2 and 3;
- by 7 if the number formed after
o cancellation of the units digit and
o subtraction of twice the value of the units digit
is divisible by 7;
- by 8 if the number formed by the last three digits is
divisible by 8;
- by 9 if the sum of its digits is divisible by 9;
- by 10 if it ends in 0;
- by 11 if the difference between the cross sums of
alternate digits is divisible by 11 (presupposes a
sufficiently high N);
- by 12 if it is divisible by 3 and 4 .
The number N = 4893 can be shown to be divisible by 7:
4893 => 489-2 • 3 = 483 = 69 • 7 .
Continuing the criterion,
48-23 = 42=>4-22 = 0, which is divisible by 7 .
The number N = 6 793 522 179 658 192 can be shown to be
divisible by 11:
6 + 9 + 5 + 2 + 7 + 6 + 8 + 9 = 52
7 + 3 + 2 + 1 + 9 + 5 + 1 + 2 = 30
A = 22 = 2 • 11

Chapter 4 CORNERSTONES OF MATHEMATICS
Sheepish 'Division
A newcomer visited a sheep station in Australia. Asf^ed to estimate
the number of sheep in the milling flocf^ she responded "10461",
almost immediately.
Stunned by the prompt and ei(act answer, the rancher as^ed how
the visitor did it.
ucIhat}s easy/ was the reply, "I just counted the hooves and
divided by four.,}
Operational Precedence
Operations of multiplication and division always take
precedence over addition and subtraction:
4 + 5x3-6/2-7 = 4 + 15-3-7 = 9
If an addition and/or subtraction is to be made first, it must be
enclosed by parentheses:
(4 + 5)x3-6/(5-2) = 9x3-6/3 = 25
We also note the left-to-right rule of precedence for
multiplication and division:
24 24
24/2x6 = --6 = 72; 24/(2x6) = -t—t = 2,
Z Z X D
but as this principle of precedence is not universally respected,
we recommend, for the first case:
(24/2)x6 = ^ 6 = 72
Where doubts might arise, the order of precedence is identified
through the use of suitable grouping notations, also known as
aggregation symbols:
( ) open parenthesis ... close parentheses
[ ] open bracket... close bracket
{ } open brace ... close brace
These notations define the order in which operations are to be
carried out within a mathematical expression. From the core
of an expression, enclosed terms are handled successively:
x = 4{25-[3x5-2(6-2)]3}
L-4—J
L_15_J L_8 1
I 7 1
■ 21 '
I 4 I
x = 4 x 4 = 16

Section 4.2 Fundamental Operations
123
GAUSS Carl Friedrich
(1777-1855)
Congruent, Modulo
Modular Arithmetic
Calculation with remainders, or residues, is convenient for
handling large numbers. This arithmetic, formalized by
Carl Friedrich Gauss, is called modular arithmetic or
arithmetic of residue classes.
We say "a is congruent to 6, modulo c", and write
a = b, mod c
when b - a is a multiple of c.
For instance,
10 = 1,mod9; 100 = 10,mod9; 1000 = 1,mod9; ...,
10 = 2, mod 8 ; 100 = 4, mod 8 ; 1000 = 0, mod 8 ; ...,
because 10-1; 100-10; 1000 - 1 and 10 - 2; 100-4; 1000-0
are all multiples of, respectively, 9 and 8.
The following calculation rules hold:
If
a\ = &i, mod c ; a<i = b<i, mod c ,
then
a\ + a<i = &i + &2 > mod c
a\-a<i = b\-b<i> mod c
a\ x a<i = b\ x &2> mod c .
• Find the least integer remainder of
(12 345 • 123 456 • 1 234 567) / 11.
Instead of multiplying out the product, replace each factor by its
remainder when divided by 11.
We have, modulo 11,
12 345 = 3 ; 123 456 = 3 ; 1 234 567 = 4,
and find
12 345 • 123 456 • 1234 567 = 3 • 3 • 4 = 36 = 3, mod 11.
The sought remainder is 3.
Casting Out Nines
This is a method of checking the results of additions,
subtractions, multiplications, and sometimes divisions.
Form single-digit sums of all terms, sums, differences,
factors, and products. Perform the same mathematical
operations as with the original terms, factors, etc., and form new
single-digit sums if required. If the underlined check digits
do not agree, there is an error.
Addition:
3 756
17 395
+ 6428
27 579
3 + 7 + 5 + 6 = 21
1 + 7 + 3 + 9 + 5 = 25
6 + 4 + 2 + 8= 20
2 + 7 + 5 + 7 + 9 = 30
2 + 1 = 3 ^
2 + 5 = 7
2 + 0 = 2
3 + 0 = 3
3 + 7 + 2 = 12
1+2 = 31
= 3

124
Chapter 4 CORNERSTONES OF MATHEMATICS
Subtraction:
37 482
- 25963
11519
Multiplication:
32756
x 8975
163780
229292
294804
262048
293985100
3 + 7 + 4 + 8 + 2 = 24
2 + 5 + 9 + 6 + 3 = 25
1 + 1 + 5 + 1 + 9 = 17
3 + 2 + 7 + 5 + 6 = 23
8 + 9 + 7 + 5 = 29
2+9+2+9+
+ 8 + 5 + 1 = 37
2 + 4 = 6
2 + 5 = 7
1+7 = 8
2 + 3
2 + 9
1 + 1
5
11
2
5x2= 10
3 + 7 = 10
6-7 =
1 + 9 =
- 1
8
8
1 + 0 = 1
>
1 + 0 = 1
j
Forming cross digit sums is equivalent to casting out
multiples of nine; a 1 in tens place, meaning 10, becomes a 1
in the digit sum and loses 10 - 1 = 9 in value; or, 10 = 1, mod 9.
To speed up the cross digit addition, cancel all the nines and
all cross digit sums equal to 9: 196 325 = 10 6S 25 = 8, mod 9.
When digit sums tally, the calculation may be correct but is
not necessarily so; the check may mask an error of nine; if
two digits have been accidentally reversed, e.g., 15 119 instead
of 11 519 in the subtraction remainder, the digit sums will tally
- but the answer is wrong.
Fractions
FOURTHS
HALVES
SIXTHS
From E.E. White, A Complete Arithmetic
Van Antwerp, Bragg & Co., Cincinnati (1870)

Section 4.2 Fundamental Operations
125
Common Fractions: p. 74
Complex Fractions
LCD
A fraction is a quotient of two quantities. Common fractions
- described in Chapter 3 - are fractions whose numerator and
denominator are both integers; complex fractions have a
fraction for the numerator or denominator or both.
To add or subtract fractions formed from integers, begin by
separating integers and proper fractions. If all fractions have
the same denominator, add the numerators according to their
signs and place the sum/difference over the common
denominator:
13
8
1
8
8 + 8
13-1-5 + 7
8
_ 14 _ 7 _ 3
"8 "4" 4
If the fractions have different denominators,
7 _5__ 2 _ _8__25 JL_II_JL
18+12~9~15"8 + 12 ~ 9 ~ 15
we must find a common denominator. The least common
denominator (LCD) is the product of the several prime
numbers occurring in the denominators, each taken with its
greatest multiplicity:
8
12
9
15
= 2x2x2
= 2 x 2 x
3
3 x3
3 x
LCD = 2x2x2x3x3x5= 360
Extend the fractions by multiplying numerator and
denominator of each fraction with factors of the LCD that are
"missing" in the denominator of that fraction:
15x45 5x30 11x40 8x24 675 + 150-440-192
8 x 45 12 x 30 9 x 40 15 x 24
360
193
360
To multiply fractions, convert all mixed numbers to improper
fractions, and place all fractions on a common bar; to reduce
the labor, cancel like factors in numerator and denominator,
and convert the result to a mixed number, if possible:
8
2 3 15 12 3 3- J • 3 • i- 3 27 3
25x4"8-5-4 " S-ii "8"38
"Invert and multiply!"
Division is equivalent to interchanging the numerator and
denominator of the devisor and multiplying by the dividend.
Convert mixed numbers to improper fractions, place on a
common bar, reduce the fraction by canceling out like factors
as before, and convert the result to a mixed number:
al I ol _ 55 I J5 _ 55 4 _ 11-^-^ _ 11 _ 5
8
8
8 ■ 15 2 i • 3 ■ i
p. 76
Conversion of fractions is described in Chapter 3.

126 Chapter 4 CORNERSTONES OF MATHEMATICS
Unit Fractions
Unit fractions are common fractions with unity for numerator
and a positive integer as denominator,
n
2 11 , 2 1 1
e'g" 5 = 3+15 ** 7 = 4+^'
The Babylonians compiled tables of unit fractions in their
sexagesimal representation in order to perform division by
use of multiplication tables.
Unit fractions were used exclusively in Egypt; all other
common fractions were written as sums of unit fractions.
Ahmes the Scribe in Egypt taught that
2__1J_J_ 9_2 1 J_
13 ~ 8 + 52 + 104 10~3 + 5 + 30
(the Egyptians had a special sign for 2/3);
and Heron of Alexandria wrote
25 _ 1 1 J_ J_
13"2 + 3+13 + 78 '
Unit fractions held their position in Europe for more than a
thousand years after Heron; Fibonacci gave
J*L_I 111 1
100 "2 +4 + 5 + 50 + 100
J*L_I I I -L
100 "2 + 4 + 5 + 25
in his book Liber abaci, from 1202, which contains tables for
converting common fractions into unit fractions.
Euclidean Algorithm
Every number greater than 1 is either a prime number or a
composite number, that is, a product of two smaller positive
GCD integers. To find the greatest common divisor (GCD) of two
numbers, we use the Euclidean algorithm, a special sequence
of divisions.
To find the GCD of two numbers, say, 102 and 30, begin by
dividing 102 by 30,
102 = 3-30 +12;
continue by dividing 30 by the remainder 12,
30 = 212 + 6,
and 12 = 2-6 + 0.
The last divisor (6) - the one that leaves zero remainder - is
then the sought GCD of 102 and 30.

Section 4.2 Fundamental Operations
127
LAMtf Gabriel A theorem by the French mathematician and engineer Gabriel
(1795-1870) Lame states that the number of steps in the Euclidean
algorithm is never greater than five times the number of digits in
the lesser of the two numbers.
One use of the Euclidean algorithm is to reduce a common
fraction, another is to convert common fractions into
continued fractions.
Reduce
11661
36 777 '
36 777 = 3 • 11661 + 1794;
11661 = 6-1794 + 897;
1 794 = 2 • 897 + 0 ; the GCD is 897.
11661 11661/897 13
36 777 " 36 777/897 " 41
Continued Fractions
All real numbers can be written as periodic or nonperiodic,
terminating or nonterminating, decimal fractions. Another
way of writing a rational number is in the form of a continued
fraction,
M 1
N = a0 + ,
CL\ +
a2 +
a3 +
CL± + ...
or, in a kind of accepted mathematical "shorthand",
N = [ao, ai, a% 0,3...],
where ao, ai, a2, «3, ... signify positive integers; consequently,
every "part fraction", say,
a2 +
a3 +
0,4. + ...
will have a value between zero and unity.
The Euclidean algorithm for finding the greatest common
denominator offers a simple method for converting common
fractions into continued fractions.
Applied to 221/41, the Euclidean algorithm yields
221 = 5 • 41 + 16, 41 = 2 • 16 + 9,
16 = 1-9 + 7, 9=17 + 2,
7 = 3-2+1, 2 = 2-1 + 0,
and we derive the expressions:

128
Chapter 4 CORNERSTONES OF MATHEMATICS
221
41
5+f =5 + 5T = 5 + -i9- = 5 + -Li
— 2 + —
16 16
= 5 + :— = 5 + :— = 5 +
2 + w
9
1
2 +
-i
2 +
-i
2 +
1 +
-i
= 5 +
2 +
1 +
= 5 +
-i
= [5,2,1,1,3,2]
Polynomials
2 +
1 +
1 +
34
Algebraic Expression
Coefficient
Variable; Constant Term
Degree of Term
Similar Terms
Monomial
Binomial Trinomial
Polynomial
Degree of Polynomial
In the algebraic expression
ax2 + bx + c
letters a and b are coefficients, that is, the numerical part of the
term; x is a variable; and c is a constant term.
The degree of a term is the sum of the exponents of the several
variables of the term; the degree of a constant term is zero;
that of x is one; of x2 or xyf two; of 5x y^z2f six.
Terms of the same degree that differ only in their coefficients,
e.g. , 3x4y3 and 5x33>4, both of degree seven, are called similar
terms.
An algebraic expression with only one term is a monomial;
one with two terms a binomial; one with three, a trinomial,
etc. A polynomial may include any number of terms; the
monomial, binomial, trinomial, etc., are just special cases of
a polynomial.
A polynomial in one variable has the general form
n -1
anxn + a
n-l
X
4- ... CL i X + Cir\ ,
where the highest exponent is the degree of the polynomial.
We distinguish between first-degree or linear polynomials,
second-degree or quadratic polynomials, third-degree or cubic
polynomials, etc.; 2%3 +y2z + z is a third-degree polynomial in
x, y, and z.
Variables are usually placed in alphabetical order and by
descending degrees:
5JC5 + 2 x2y3 + 3x%yz + z4 - x2 + y2 + x - 2z
| 5th deg. | | 4th deg. | |2nddeg.| |lstdeg.|

Section 4.2 Fundamental Operations
129
Addition and Subtraction
After grouping like terms together, addition and subtraction
are carried out within each group of the polynomials.
4jc3 + 8x2y + 2xy2 - 2y3 -x2 - Iz
+ 2x2y - z2 - z + 3
4jc3 + 10jc23/ + 2xy2 - 2ys -x2 - z2 - 8z + 3
and
4jc3 + 8x2y + 2xy2 - 2y3 -x2 - 1 z
-( +2x2y -z2 - z +3)
4jc3 + 6x2y + 2xy2 - 2y3 -x2 + z2 - 6z - 3
Multiplication
The product of two polynomials is the sum of all individual
products of every term in one polynomial multiplied by every
term in the other.
To find the product of more than two polynomials, begin by
multiplying two of them, then the result and the next
polynomial, etc.:
(3jc2 + 2jc + 3)(2jc-4)(jc + 2)
= (3jc2 + 2jc + 3)(2jc2 - 8),
3jc2 + 2x + 3
x 2*2 - 8
-24jc2 - 16 jc -24
6 jc4 + 4 jc3 + 6 x2
6X4 + 4X3- 18jc2- 16x -24
Division
When dividing one integer by another, the division is carried
on until the remainder is zero, or less than the divisor.
Analogously, the division of one polynomial by another is
continued until the remainder has as low a degree as possible.
To perform the division, start by organizing the terms of both
polynomials in descending order of powers, and insert a
coefficient zero for all missing powers.

130
Chapter 4 CORNERSTONES OF MATHEMATICS
To divide
a3+2a2b+2ab2+b3
a + b
we have
a2+ab+b2
(a + b) \a3 + 2a2b + 2ab2 + b3
a3 +
a2b
a2b+ lab2
a2b + ab2
ab2 + b3
ab2 + bs
0
and
*4-18
x2-x-2
jc2 + jc + 3
(x2-x-2) |jc4 + 0jc3 + 0jc2 + 0jc-18
jc 4 - x3 - 2 x2
X3
X3
+
2x2
■ X2
3x2
3x2
—
+
—
2x
2x
3 x
-18
-6
Remainder:
5x -12
- Divide the first term of the dividend by the first term of the
divisor; enter the result as first term of the quotient.
- Multiply the divisor by the term found; enter the product
under the dividend, like terms under each other; subtract.
- Move down as many terms as required in order to continue
the operation.
- Go on until the remainder is of as low a degree as possible,
or zero.
The method of division of polynomials with real and complex
coefficients is the same.
Partial Fractions
Algebraic expressions containing a polynomial in a single
variable in the denominator, or in the denominator and
numerator, may be split into partial fractions for easier
handling.
o If the numerator is of equal or higher degree than the
denominator, divide the denominator into the numerator
until the degree of the remainder is less than that of the
denominator.
o Resolve the denominator into prime factors; prime factors
higher than degree one are permitted only if they cannot be
factored further.
o Expand the original fraction into partial fractions:
Ai A2 A
(a x + b)n
(ax2 + bx + 071
n
ax + b (ax + b)2
Ai x + Bi
a x2 + b x + c
(ax + b)n
Anx + Bn
(a x2 + b x + c)n
o
Determine the coefficients of the numerators.

Section 4.2 Fundamental Operations 131
Example:
x2 + x + 1 x2 + x + 1 A B Cx + D
= — + r +
2x4+xs + 2x2+x x(2x + l)(x2 + 1) x 2x + l ^2 + 1
To determine A, 5, C, and Z), multiply both members of the
equation by x (2 x + 1) (jc2 + 1):
jc2 +jc +1 = A (2 x + 1) (jc2 + 1) + Bx (x2 + 1) + (Cx + D) x (2x + 1)
Or^ + lj^+lx + l = (2A + B + 2C)x3 + (A + C + 2D)x2 + (2A + B + D)x+A
Equate coefficients of like powers of x :
>
2A + 5 + 2C= (n
A + C + 2D = 1
2A + B + D= 1
A = 1
and
jc2+jc + 1 1 6 1 2jc
>/
A= 1
5 = -6/5
C = -2/5
D = 175
2jc4+jc3 + 2jc2+jc x 5(2x+l) 5 (x2 + 1) 5 (x2 + 1) '
p. 745 This technique is used in finding integrals and in solving
differential equations.
Complex Numbers
p. 87 The sum of complex numbers is obtained by adding their real
parts and their imaginary parts, separately; for instance,
(5 + 3i) + (3-2i) = (5 + 3) + (3-2)i = 8 + i;
their difference is, analogously,
(5 + 3i) - (3-2i) = (5-3) + (3 + 2)i = 2 + 5i .
The product of two complex numbers is found by multiplying
each term of the one by every term of the other, remembering
that i2 = -l,
(5 + 3i) x (3-2i) = (15-6i2) + (9-10)i = 21 -i.
Conjugate Complex Numbers A special case is offered by conjugate complex numbers
Notations z and z* are equivalent, z = a + b i and z * = a - b i, where a and b are assumed to be real
numbers; their product is always real,
22* = (a + bi) (a - b i) = a2 - b2 i2 = a2 + b2;
for instance,
(3 + 2i) (3-2i) = 9 + 4 = 13.
The quotient of two complex numbers is obtained by
multiplying numerator and denominator by the conjugate of the latter;
thus,
5 + 3i _ (5 + 3i) (3 + 2i) 9 +19i 9 19 .
3-2i"(3-2i)(3 + 2i)" 13 " 13 + 13 X"

132 Chapter 4 CORNERSTONES OF MATHEMATICS
4.3 Laws of Arithmetic and Algebra
Definitions and rules that will be discussed here are valid for
ordinary mathematical operations but do not all apply to
special algebras - e.g., Boolean algebra and algebra of trans-
finite numbers (Chapter 7), vector algebra (Chapter 16), and
matrix algebra (Chapter 18).
o For every real number a there is a real number -a, called
the additive inverse or the opposite or negative of a, although
it is not necessarily itself a negative real number; for
instance, if a =-3, then -a = 3.
o For every real number a, a real number 1/a exists,
commonly called the "inverse of a" or the "reciprocal of a",
instead of the longer "multiplicative inverse of a". Since
division by 0 is not possible, \la exists only if a * 0.
The following laws are not likely to surprise anyone; the
surprise is if the laws do not apply. The incentive to formulate
these laws came from discoveries that seemingly self-evident
rules did not apply to certain operations in new branches of
mathematics.
Basic Laws of Identity
and Equality
Reflexive law a = a
Law of symmetry If a = b, then b = a
Transitive law If a = b and b = c, thena=c
Substitution law If a = b, then b can replace a in
any equation
Definition of | a \
Identity laws of
Addition and
Multiplication
Law of additive inverse
Law of multiplicative
inverse
\a\ = aif a>0
\a\ =-aifa<0
a + 0 = 0 + a =a
a • 1 = 1 a =a
a + (-a) = (-a) + a = 0
11 n %P
a • — = — • a = 1, if a * 0
a a '
Commutative laws of
Addition and a + b = b+a
Multiplication ab = ba
In a commutative system, the order of adding or multiplying
terms is inconsequential; in a non-commutative system the
order is crucial (a + b*b + a;ab*ba), as in multiplication of
vectors and matrices.
In group theory, a group following the commutative laws is an
ABEL Niels Henrik abelian group, named after Niels Henrik Abel; abelian
(1802 -1829) groups are of central importance in branches of modern
mathematics, notably algebraic topology.

Section 4.3 Laws of Arithmetic and Algebra 133
Distributive law of
Addition and
Multiplication a(b + c) = ab +ac
In commutative operations, a product can be rendered as a sum
and vice versa.
For non-commutative operations, we distinguish between
left distributivity, a(b + c) = a b + ac ,
an right distributivity, (b + c) a = b a + c a.
Associative laws of
Addition and a + (b + c) = (a + b) + c
Multiplication a (be) = (a b)c
In systems where the associative laws apply, any method of
grouping may be used, that is, at any stage of addition (or
multiplication) one may add (or multiply) two adjacent terms.
The commutative and distributive laws were formulated by
Francois-Joseph Servois, the associative laws by William
Rowan Hamilton.
From the above, we may derive several other laws and
definitions for real numbers:
Multiplication law of zero 1. ax0 = 0xa = 0
The zero product law 2. If a b = 0, then either a = 0
or 6 = 0
Laws of negation 1. - (-a) = a
2. (-a)b = a(-b) = -(ab)
3. (-a) (-b) = ab
Cancellation laws of
Addition If a +x -a +y, then* =y
Subtraction If a -x = a -y, then x = y
Multiplication If a * 0 and a x = a y, then x = y
Addition and subtraction
laws of absolute values |a±6| < \a\ + \b\
Multiplication law of | a 6 | = | a | | 6 |
absolute values
Division law of If b * 0, then
absolute values
a
b
a
b
(Division Jattacy
To attempt to prove that 1 = 2, let x - 1 and x = y,
multiply both sides by y: 0
xy = yz\
subtract*2, 9 n 9
xy -jcz = yz - jcz;
factor> / X / w \
x (y -x) = (x +y) (y -x);
and, finally, divide by (y - x),
x - x +y .
In every mathematical game lies a trap; here, (y - x) is equal
to zero and cannot be used as a divisor.

134
Chapter 4 CORNERSTONES OF MATHEMATICS
4.4 Powers and Roots
History
ARITHMETI
CA INTEGRA
Audiore Michaele Stifdic
Cumpnefarione Philippi Mclanchthoni*.
NorimbcrgT apud Iohan. Pordum.
Anno Chrifti m. x>. xiiiu.
Cum gratia & priuilegi'o Cxfarco
aces Regio ad Scxennium*
STIFEL Michael
(c. 1487 - c. 1567)
Powers
A product ofp equal factors a • a • a • ... a • a = a?, where a is the
base andp the exponent, is called thep-th power of a and reads
"a raised to thep-th power".
The concept of powers was known to the old Babylonians
and Egyptians; in the Rhind Papyrus, Ahmes the Scribe used
a word meaning mass or quantity to denote the unknown
quantity that we call x.
In the days of rhetorical mathematics, powers of the unknown
had to have names before they could be described by symbols.
The ancient Greeks, to whom mathematics meant geometry,
called the square of the unknown a tetragon number, that is, a
four-corner number.
Diophantos of Alexandria used the word power (Greek:
dynamis) for the square of the unknown; the third power was
a cube; the fourth, a power-power; the fifth, a power-cube; and
the sixth, a cube-cube.
To Arab writers, the unknown was shaif which means thing or
anything; its square was mal, meaning wealth. The early
Latin writers took over these terms as res meaning thing, and
census for the evaluation of wealth, or tax. Algebra became
Ars rei or Ars rei et census.
Italian writers translated res as cosa, which became die Coss
in German and, in English, cossike arte.
The first algebraic power symbols corresponding to our x, x2,
x3, x4, etc., to appear in print were found in the Arithmetica
integra by the German mathematician Michael Stifel, to be
followed by symbols invented by other writers:
1544
1572
1585
1591
1631
1634
1637
Stifel
Bombelli
Stevin
Viete (Vieta)
Harriot
H6rigone
Descartes
A
X
®
A
a
a
a
AA
&
®
Aq
aa
a2
a2
AAA
vS,
(D
Acu
aaa
a3
a3
AAAA
4,
®
Aqq
aaaa
a4
a4
AAAAA
^
®
Aqcu
aaaaa
a5
a*
The symbols used by Descartes agree with those used today,
except that he recognized only positive integers as exponents;
the introduction and acceptance of negative and arbitrary real
exponents is the work of John Wallis and Isaac Newton, and
imaginary and complex exponents are mainly Euler's work.

Section 4.4 Powers and Roots
135
Roots
Py-
Thep-th root of a number a, ya , is a quantity that, raised to the
p-th power, gives a; that is, it satisfies the relation
(Va) = a .
Like powers, the roots of numbers have a long history; the
Egyptians knew how to calculate the square roots of numbers,
including simple fractions, as early as about 4000 years ago.
Arab writers conceived of a square number as grown out of a
root, while Greek and Latin scholars thought of roots as the
sides of a geometrical square. Thus, when mathematics was
still rhetorical, works translated from Arabic generally used
the word radix for "root", while those derived from Greek and
Latin had latus for "side".
The symbol most commonly employed by late medieval
writers to denote a root was R - a contraction of the word radix
embodying its first and last letters - which, with numerous
variations, held its place in manuscripts and printed books for
more than a century:
4th root
R4
R-4a
1484
1494
1521
1539
1572
Chuquet
Pacioli
Ghaligai
Cardano
Bombelli
sq.
R
R.
R
R
R
root
R2
■ 2a
□
. q.
CU. ]
R3
R
R
R
root
i
• 3a
□
. cu
R . c.
The modern symbol y first appeared in print in 1525 in
Christoff Rudolff s Die Coss ("Algebra"):
sq. root cu. root 4th root
1525 Rudolff V~ CVT" W~
1553 Stifel Z/ $/ M/
1628 VI acq V~ Vc~ VT~
1659 Rahn V~ <® V~V~; V~

1 36 Chapter 4 CORNERSTONES OF MATHEMATICS
Calculation Rules for Powers
The notation aP means the product of p factors a,
a.a.a...a.a = a,P)
Base, Exponent, Power where a is the base and p the exponent of the power aP.
The exponent p was originally assumed to be always a positive
integer, but has since been generalized to include also
negative integers, zero, arbitrary real numbers, and
imaginary and complex numbers.
A product of, say, six factors of 2 may be written
22222.2 = 26 = 64.
The first power of a number is the number itself, 21 = 2; the
second power is its square, 22 = 4; the third power is the cube,
23 = 8; etc.
The multiplication of powers of the same base,
a .a . a . ...a .a x a -a -a - ...a -a = am . an = am + n
{m factors) (n factors)
gives the rule:
o To multiply powers of the same base, add their exponents.
Division of powers of the same base,
(m factors) a . a . a . ... a . a am
= _ am - n
(n factors) a . a . a . ... a an
provided that a * 0 ; that is:
o To divide powers of the same base, subtract their exponents.
This rule has two corollaries,
am
_ = am-m = a0 = i (a .* 0)
am
and
a° 1
= arn (a * 0).
an an
The 7i-th power of the ra-th power of a is
{a > a > a > ... a)x(a • a . a . ... a) x ... x (a . a . a . ... a)
(m factors) (m factors) (m factors)
n factor parentheses
= (am)n = amn
and, analogously,
(am)1/n = am/n .
am -bm = (a b)m : am I bm = (a/b)m (b * 0).

Section 4.4 Powers and Roots
137
pp. 791 et seq.; pp. 793 et seq.
p. 123
Slightly more complicated power expressions are simplified
thus,
(b-a)n + 1 (a-b)n + 1
(a-b)2n + 4 (a-b)2n + 4
(-1)
n+ 1
= (-1)»+1 (a-b)-(n + V .
Powers of imaginary and complex numbers are defined in the
same manner as powers of real numbers,
7i factors
and obey the same rules.
Positive integer powers of the imaginary unit i repeat in cycles
of four,
il = i
i2 = -l
i3 = -i
i4=l
I5 = i
i6=-l
i7 = -i
i»=l
Since a0 = 1 if a * 0, we have
i° =
i9 = i
iio = -l
in = -i
112=1
= 1.
Negative integer powers of i are
i-2 = J- =
. -1; i"3 =
i13 = i
il4=-l
i I5 = -i
116=1
ad infinitum.
i
— = +i ; etc.
Powers and roots of complex numbers, and their products and
quotients, will be described in Chapter 24.
Arithmetic of residue classes was discussed in Section 4.2.
For powers, we have: If
a = 6, mod m ; n is a positive integer,
then an = bnt mod m.
Find the least integer remainder of 3263 divided by 6567.
The 126 digits of the expansion of 3263 are too many to handle
for most pocket calculators.
Use the following algorithm, in modulo 6567:
32 = 9
= 81
34 = (32)2
38 = (34)2
316 = (38)2
= 6561 =-6
= (- 6)2 = 36
332 = (316)2 = 362 = 1296
364 = (332)2
3128 = (364)2
= 1296" = 1 679 616 = 255 • 6567 + 5031 = 5031
= 50312 = 3854 • 6567 + 1743 = 1743
3256 = ¢128)2 s i7432 = 462 . 6567 + 4095 = 4095,
which is as close as we can get to 3263 by repeated squaring.

138
Chapter 4 CORNERSTONES OF MATHEMATICS
Continuing in modulo 6567, we find
3263 = 3256 . 37 = 3256 . 34. 32. 3 = 4095.31 . 9.3
= 8955 765 = 1363 • 6567 + 4944 = 4944.
The sought remainder is 4944 .
Calculation Rules for Roots
Radical, Radicand, Root Index
There are n distinct n-th roots of a number N * 0. Most of these
will be complex numbers. They satisfy rn = N and might be
written
where ^jN is the radical, N is the radicand, and n is the root
index. More generally, we may allow the root to be real,
imaginary, or complex.
When the alternative symbol v - without the bar - is used on
a composite radicand, this must be enclosed in parentheses,
Va + 6 = V (a + b).
Second roots, Vp, are called square roots, and are written
without the root index; third roots, yp, are cube roots.
Principal Roots, Secondary Roots We distinguish between principal roots and secondary roots:
0 The principal root of a positive number is the positive root;
other roots, of which there are n - 1, are secondary roots:
V4 = +2; -2 is a secondary root;
\1 = +1; -1, +i, and -i are secondary roots.
0 The principal n-th odd-index root of a negative number is,
by definition, the negative real root.
3/—
Example: V -8 has the principal root -2; the two other roots,
(l + iV3)and(l-iV3), are secondary roots.
Check: (l + iV3)3 = -8.
0 Negative numbers have no even-index principal roots; the
square roots of-4 are +2i and -2i; neither is a real number
and, hence, no principal root.
Using the power notation of radicals,
^77 = NVn (n * 0),
we have the following calculation formulas for principal roots
of a, b > 0:
m
o fa . VT = V ab = (a b)Vn
n,— , n,— rii /aV^1
o \c /\6=\a/6= 7
m I
o \ ^sfa = m\a = a1/mn
m.
mi /71 /All— .
o yan = V cl = an/m

Section 4.4 Powers and Roots
139
Surds
Latin, surdus, "deaf, "silent",
"indistinct"
Conjugate Binomial Surds
Simplified Form
Rationalizing the Denominator
A radical expressing an irrational number is called a surd,
qualified as quadratic (e.g.,V~2), cubic (V2), quartic (\2),
quintic Cv2), etc., after the index of the radical.
(The term surd is sometimes used as a synonym for irrational
number.)
A pure surd, or entire surd, contains no rational number, that
is, all its factors or terms are surds, e.g., y2 or y2 + *y3.
A mixed surd contains at least one rational term, 2 + V~3, or
factor, 3 ^2.
In a binomial surd, at least one of the numbers must be a surd,
2 + V2 or V2 + V3; in a trinomial surd, at least two terms must
be surds that cannot be reduced to a single surd, e.g.,
2 +V2 +V3 orV2 +V3 + V5.
Conjugate binomial surds differ in the sign of one of the
irrational terms; the product of two conjugate binomial surds
is rational; e.g.,
(2+V2M2-V2) = 2 ; (V2 + V3) (V2 - V3) = -1;
(V^ + v&)
V^ - V& (V^ - V&) (Va + V&)
a - b
A radical is said to be in simplified form if its radicand is not
in fractional form and does not include powers of the same
index as the root index:
V72 = V2x2x2x3x3 = 2x3V2 = 6^
The process of eliminating surds from the denominator is
called to rationalize the denominator or, simply, to
rationalize.
V3
V3 (2 V5 + V3)
V3 (2 V5 + V3)
2V5-V3"(2V5-V3)(2V5 + V5"=(2V5)2-(V3)2
2^15 + 3 1
20-3
= —(3 + 2V15) .
Simplification and rationalization were essential skills when
roots had to be found in tables and computations done by paper
and pencil alone. They still help us understand how to study
the nature of specific numbers or expressions.

140
Chapter 4 CORNERSTONES OF MATHEMATICS
The Binomial Theorem
!: p. 106
(?) : P- **
Pascal's Triangle
Integral powers of the binomial (a + b) are
(a + 6)0
(a + 6)1
(a + b)2
(a + 6)3
(a + b)4
= 1
la +16
la2 + 2a6 + 162
la3 + 3a2b + 3a&2 + 163
la* + 4a36 + 6a2b2 + 4a63 + 164.
Higher powers are simply expressed by using the binomial
theorem, a way of expressing the /i-th power of the binomial
(a + b) in symbolic form:
(a + 6)» = (*)a» + (n1)an~1b + (^a""^2
+ ... + (/^)^-1 +0 6-
or, abbreviated,
Al
;a + &)* = ^(^)a*-^ ,
; = o
where the coefficient of an ~l b l is the number of ways we can
chose b's from i of the factors; the other (n - i) factors contribute
a's. The formula is
(n) ±1
If we place the coefficients of the terms in the binomial
expansions in a triangular array, with (a + 6)0 at the top, the
coefficients of (a + 6)1 in the second row, those of
(a + b)2 in the third row, etc., we obtain Pascal's triangle.
n =
0
1
2
3
4
5
1
1 1
12 1
13 3 1
14 6 4 1
1 5 10 10 5 1
6 15 20 15 6 1
7 21 35 35 21 7
Every coefficient is the sum of the two coefficients straddling it
in the row immediately above,
n
n
(n-i)\ i ! [n - (i + 1)] ! (i + 1) !
n ! [(¾ + 1) + (/1- i)] (n + 1) !
(n-i)\ (i + 1) !
(/i-i) ! (i + 1) ! '

Section 4.4 Powers and Roots
141
PASCAL Blaise
(1623-1662)
CHU Shih-chieh
p. 51
or, in symbolic notation,
0)-6:0-(::1)-
Pascal's triangle is named after the French mathematician-
philosopher-physicist Blaise Pascal, who described it in his
posthumously published Traite du triangle arithmetique
(1665). Although Pascal never claimed recognition for his
discovery, his name is inseparably linked with it.
The triangle is, in fact, much older; it appeared as early as
1303 in Precious Mirror of the Four Elements by the Chinese
mathematician Chu Shih-chieh.
The chart uses Chinese rod numerals.
•5K.<x nys yarolldti
i^/riana le .Ant'limeliaue
Fig. 9
Le triangle arithmetique
de Pascal
From Pascal's Traite du triangle
arithmetique (1665).
"Chu Shih-chieh's Triangle" (1303). From
Joseph Needham, Science and Civilization in
China III: Mathematics and the Sciences of
Heavens and the Earth (Cambridge
University Press, 1968).
Root Extraction
Square Roots
Before the time of the electronic calculator, square roots of real
numbers were generally determined with the help of tables of
squares or square roots. If these helpmates are not available,
we can still find the square root of a real number by direct
calculation.

142
Chapter 4 CORNERSTONES OF MATHEMATICS
Dividing and Averaging
The following simple method to find square roots of a real
number goes back to the time of the ancient Babylonians.
If
a better approximation is
*Jn
N/n
-h
n
N
and increasing accuracy is obtained as we continue:
<N * 5
iV
1 /AT
2 7T+n
1 /W
<
N
1
2
JV
1 /2V "\ 1 /W
2 [n+n +2 7T+"
e£c.
1
2
JV
1 /2V
1 (N
2 U
+ 2
71
+ n
+ n
>
Sought: VlOOO
A rough approximation (first guess) is
VlOOO * 30.
Continuing the approximations, we find:
VlOOO * ^(^- + 30 I = 31.666 666 66..
2 I 30
VlOOO * ^
1000
2 I 31.666 666 66
Viooo * ^
1000
2 31.622 807 01
Viooo * ^
1000
2 131.622 776 60
+ 31.666 666 66 = 31.622 807 01
+ 31.622 807 01 = 31.622 776 60
+ 31.622 776 60 = 31.622 776 60
Thus, from the initial guess, we achieve a result correct to
eight decimal places in just three steps.

Section 4.4 Powers and Roots
143
Continued Fractions
p. 127 To extend the method of continued fractions to irrational
numbers, we choose \6, which lies between 2.4 and 2.5, since
2.42 = 5.76 and 2.52 = 6.25.
We can write
V6 = &o + = K>0&1 &2 &3--J- (1)
b1 +
b2 +
63 + ...
Since the fraction must be less than unity, we let 6q = 2, and
obtain
= ^6 - 2 (2)
[b\ b<i 63 ...]
or
[bl 62 63-.I = h + [6263\4j = -^-^ . (3)
Letting b\ = 2, we have
1 -2=%* (4)
[62 &3 &4---1 V6- 2
and
1 2 t-
[62 &3 &4--J = 62 + fa a h T = T= = V6 + 2. (5)
L63 64 65...] ^/6-2
With 62 = 4, we obtain
r, A T = ^6 + 2-4 = V6-2, (6)
I03 04 05 .. .J
that is, we have the same relation in Eq. (6) that we had in
Eq. (2); hence, the pattern repeats:
63 = 2; 64 = 4, etc.
We can now write
V6 = 2 + —^
2 +
4 +
2 + —^
4+ 1
2+...
or with the notation introduced for periodic decimal fractions,
V6 = [2, 2, 4, 2, 4...] = [2, 2, 4 J .

144 Chapter 4 CORNERSTONES OF MATHEMATICS
Many algebraic irrational numbers produce relatively simple
periodic continued fractions. A number's continued fraction
is periodic precisely when the number is a quadratic surd:
V2 =1 + ^ = [1, 2, 2, 2, 2...] = [1, 2]
2+:—5—
2 +
2+ *
2 +
V3 = l + = [1,1,2]
1 + :
2 +
i+—*
2+ 1
1 + ...
VTI = 3 + : = [3,3,6]
3+;—~
6 +
3+ X
6 + ...
Vl7 = [4, 8]
p. 776 Continued fractions converge more rapidly than power series
expansions and are often useful means for finding
approximative numerical values of irrational numbers.
Method for Real Numbers and Polynomials
We are required to find the square root of 9467.29 .
Step 1. Begin by dividing the radicand into groups of two
digits, working both ways from the decimal point.
Use a sign f to separate the groups.
9 a/94'67/29 = 97.3
+ 9 9x9=81 31
187 13 67
+ 1 187x7 = 1309 13 09
1943 58 29
3 1943x3=5829 58 29
0
Step 2. Find the largest number whose square is less than the
leading two-digit group (94) of the radicand; place
this digit (9) in the root and in two positions on
the left, in front of the extraction galley, and add these
two digits (9 + 9 = 18).
Step 3. Calculate the square (9 x 9 = 81) of the first root digit,
and subtract it from the first group (94) of digits in the
radicand, leaving a remainder (13).

Section 4.4 Powers and Roots
145
a
+ a
2a + b
b
2(a + b) + c
c
Step 4. Move the next two-digit group (67) down from the
radicand to form a new minuend (1367). Assess the
next digit of the root, place it (7) in two positions on the
left as before.
Step 5. Carry out the addition (187 + 7 = 194) and the
multiplication (187 x 7), and introduce the product (1309)
in the galley (under 1367); subtract to obtain the
remainder (58).
Step 6. Move the last two-digit group (29) down from the
radicand to obtain a last minuend (5829). Determine
the next digit of the the root (3), add it on the left,
and perform the multiplication (1943 x 3) and place
the product (5829) in the galley; subtract to find the
remainder, zero.
To prove the correctness of this extraction, we let
a = 90
6=7
c= 0.3
and find
a x a =
(2a + b)b =
\/a2 + b<l + c2 + 2 a b + 2ac + 2 b c
1 2
= a + b + c
_a
0
b2
b2
2ab
2ab
0
0
[2(a + b) + c ] c =
2ac + 2bc
2ac + 2bc
0
0
0
23
Any fool can touch a
Sutton, fmt...
Calculate V2 to five correct decimal places - a most tedious job
that might make us appreciate the tenacity of the constructors
of the square root tables.
1
+1
24
+_^
281
+ 1
2824
+ 1
28282
+ 2
282841
+ 1
^.'OO'OO'OO'OO'OO = 1.414213..,
1x1=1 = 1.41421
1 00
24 x 4 = QS.
4 00
1 19 00
1 12 96
6 04 00
5 65 64
38 36 00
282841 x 1 = 28 28 41
2828423 10 06 59 00
3 282842 x 3 = 8 48 52 69
1 58 06 31
281 x 1 =
2824 x 4 =
28282 x 2 =

146 Chapter 4 CORNERSTONES OF MATHEMATICS
Using this method for a polynomial, we begin by arranging
all terms by descending degrees.
x2
V^4-2jc3 + 3jc2-2jc + 1 = x2-x+l
X2 X4
2x2 -x -2jc3 + 3jc2
- x - 2 jc3 + x2
2x2-2x + 1 2x2 - 2x +1
1 2x2- 2x +1
0
- Find the square root of the first term of the radicand; enter
it in the quotient and twice in the control column to the left of
the radical.
- Square the term and enter the result under the first term of
the radicand; subtract.
- Add the terms in front of the radical to get the sum 2 x2.
- Move the next two terms of the radicand down; divide by
2 x2 to find the next term, (- x ), of the quotient.
- Add (-x) as shown, multiply as before, and place under the
dividend; subtract.
- Move the remaining two terms of the radicand down, find
the last term of the quotient, +1, and complete the
multiplication; subtract.
Cube Roots
Method of Trenchant
The following method of extracting cube roots was used by the
TRENCHANT Jean French mathematician and physician Jean Trenchant in his
treatise Arithmetique (1557), but is probably older.
V 100 441020 = 464.8
(4) 64
36 441
4800 (6) 33336
3105 020
634 800 (4) 2561344
543 676 000
64588800 (8) 517601792
26 074208 000
etc.
We have found the good physician Trenchant's method an
excellent cure for insomnia:
o Begin by dividing the number into groups of three digits,
working from the last to the first digit of a whole number or,
if a decimal fraction, in both directions from the decimal
point. Here: 100 441020

Section 4.4 Powers and Roots
147
o Find the root of the greatest possible cube contained in the
first group of three digits,
43 = 64 < 100 .
o Subtract the cube, 64, from 100 to obtain the remainder 36.
o Move down the next group of three digits, 441, to give a new
dividend, 36 441.
o To find the second digit of the cube root, keeping in mind
that shifting a digit one step toward the left means a tenfold
increase of its value, we write a = 40 and assume the second
digit to be b.
o The dividend 36 441 must then contain the sum 3a2b + 3ab?
+ 63; disregarding the smaller terms, we compute
3a2 = 3.402 = 4800
and place it on the left of the extraction.
o We find that this divisor is contained 7 times in the
dividend 36 441.
o Compute the terms
Sa2b = 3-402-7 = 4800- 7 = 33600
Sab2 = 3-40-72 = 120-49 = 5880
63 = 73 = 343
and find the sum 39 823,
which, however, exceeds 36 441. The method then requires
that one tries with the preceding, smaller, integer (b = 6);
thus: 3 • 402 • 6 + 3 • 40 • 62 + 63 = 33 336, which we now enter
below 36 441 and perform the subtraction to find the
reminder 3105.
o Move down the next group of three digits, 020, to give the
dividend, 3 105 020.
o If we now let 460 = c, we have 3 • c2 = 3 • 4602 = 634 800,
which we enter as the next divisor on the left.
o Division of 3 105 020 by 634 800 gives the next digit of the cube
root, d = 4.
o We now compute
3c 2d = 3-4602-4 = 2539200
3c d2 = 3-460 • 42 = 22 080
ds = 43 = 64
and the sum 2 561 344
which we place under 3 105 020 and subtract to find the new
remainder 543 676.
o Next, add as many groups of three zeros as you desire
further decimals in the cube root, and move down the first of
these groups to give the dividend 543 676 000.
o Continue the calculation and extraction in the manner
described above.

Chapter 4 CORNERSTONES OF MATHEMATICS
o We find the next divisor 3 • e2 = 3 - 46402 = 64 588 800, which
is contained f = 8 times in the dividend.
o Now compute
Se2f = 3-46402-8 =516 710 400
Sef2 = 3-4640- 82 = 890 880
f3 = 83 = 512
and the sum 517 601 792
etc.
Method of Heron
The following method of calculating cube roots was used by
Heron of Alexandria (1st century) but is much older, probably
dating from the time of the ancient Babylonians.
Sought: n = yN
Enclose n between the consecutive integers a and 6,
a<n<b ; a + 1 = b
where n-a = p; b-n = q\ p + q = 1.
a3 N 63
->
a n
We then have
a3 = (n -p)3 = n3-3n2p+ Snp2-p3
63 = (n + qft = n3 + 3 n2 q + 3 n q2 + q3
Since
p < 1 ; p3 « 1 and q < 1 ; q3 « 1,
we discard the terms p3 and q3, and obtain
P = N-a3 & Snp(n-p) = Sanp
Q = b3-N & 3nq(n+p) = Sbnq
and
p b P p b P
— ^ —TT \ = P ** ",—t: 7? •
q a • Q P + q b • P + a • Q
We can now write
b P
n =a + p & a + -— — .
y b-P+a-Q
Example: N = 90; a = 4, a3 = 64; 6 = 5, 63 = 125
P= 90-64 = 26
Q = 125-90 = 35
5 • 26 130
n & 4 + -———-—— = 4 + -=rr = 4.481
5-26 + 4-35 270
which is correct to three decimal places.

Section 4.4 Powers and Roots
149
Sought: V400
We have
a = 7
6 = 8
a3 = 343]
63 = 512
P = 400-343= 57
Q = 512-400 = 112
b-P = 456
a • Q = 784
b-P + a-Q = 1240
456
rc ** 7 + 7^77 = 7.3677...
1240
(The correct value of the cube root is
^400 = 7.368 062....)
Other Methods
There are several methods of approximating the cube root of a
number N = a3 + b :
o Hindu mathematicians used the approximation,
Va3 + fe ** a +
3 a2
as early as the 5th century;
o Tartaglia (c. 1500 - 1557) used a closer approximation,
Va3 + fe ** a +
3 a2 + 3 a '
which can also be found in some 15th-century manuscripts;
o Fibonacci (c. 1170-c. 1250) used
cf. pp. 775 et seq.
Va3 + 6 ** a +
3 a2 + 3 a + 1 '
Through the use of calculus, we have even more powerful
methods for finding cubic and other roots.
The above methods all work as well as one might want. We
make a our best current approximation, and b the remainder;
using Newton's binomial series, we have
<l
o3fl +
M b\M / 1 b
= a 1 + —r I = a 1 + — • —-+ ...
a
a
a
of which the ancient Hindu mathematicians had an
approximation of the first term.

Chapter 4 CORNERSTONES OF MATHEMATICS
Logarithms
History
The rebirth of science after the state of rigor mortis imposed
by the Church in the Middle Ages was particularly evident
in the awakened interest in astronomy and in the attendant
development of trigonometry, generated also by other forms
of world exploration - land surveying, cartography, and
navigation.
Scientists everywhere began to spend enormous amounts of
time in calculating tables of trigonometric functions, and
it became important to find methods of replacing the often
laborious operations of multiplication and division with addition
and subtraction, -e.g., by employing formulas such as
2 • sin a sin /3 = cos (a- /3) - cos (a + /3) .
John Napier, laird of Merchiston, near Edinburgh, Scotland,
voiced his opinion thus:
There is nothing more troublesome in mathematics than the
multiplications, divisions, square and cubic root extractions
of great numbers which involve a tedious expenditure of
time, as well as being subject to "slippery errors".
Napier presumably had in mind a passage in Michael StifePs
Arithmetica integra (1544) where he compares a sequence of
consecutive integer numbers with a sequence of corresponding
powers of 2 having these integers as exponents:
0123456789 10
1 2 4 8 16 3264128256 512 1024
From this it is evident that sums and differences of the power
indices correspond to products and quotients of the powers
themselves - but equally that the number 2 as a base would be
impracticable for purposes of computation because of the large
gaps between successive terms, which would make
interpolation altogether too inaccurate. One would have to choose a
number much closer to 1.
Napier's stated aim was to facilitate the calculation of natural
sine e sines and other trigonometric functions. In Napier's time, the
sine of an arc, or an angle, was not thought of as a ratio, but
rather as half the length of the chord subtending the double
angle at the center of a circle of a given radius - actually, of
course, the ratio of this semichord to the radius of the circle.
To avoid introducing fractions, Napier chose 107 as a value for
the radius of the circle. As base for his powers, he decided on
(1-10-7) = 0.999 999 9.
By multiplying all powers by 107, we have the relation
N = 107(1-10-7)L,
where L is the "Naperian logarithm" of the number N.

Section 4.5 Logarithms
1
373 A
DESCRIPTION
OF THE ADMIRABLE
TABLE OE
LOGARITHM ES:
WITH
A DECLARATION OF
ThiMost Plbntifvl,Easy,
and (needy vie thctcof inbothkindcs
of Ttigonomcttic, as alio in ill
Mathematicall calculations.
INVENTED AND PVBLI-
suid In Latin By That
Honorable 1.. lo'lM JS|Jvpair,Bi-
ron of Mmb)JJo»,and traniUtco into
EnglHH by tUc late leatned and
famous Mathematician
Edward Wright.
Wkh an Addition of >an Inflrnmcntall Tabic
to finde the f**s proportionally indented by
^$bcTvd»(Ut6r} and dcjlribed in the end
of the *o*\e by Hbn&y Biua*
Geometry reader at GuO&mr
l*«fcinLondtH.
2 Thefirft Bceke. Chapj
A to C, In the fecond moment ftom C to D.
In the thitd moment ftom D to E,& Co fotth
infinitely,dcfctibing the line A C D E F,&c.
The fpaces-A C, C D, D E, E F, &c. And all
the teft being eqaall, and defctibed in equall
moments(ot times.)This line by the formet
definition (hall be fai4 to increafe equally.
A CorelUry Therefore by this increafmg% quantities equally
or confe
quent
differingjnuflntedes be producedfn times
equally differing.
As in the Figute before,B went fotwatd
ftom A to C in one mom en c, and from A to
£ in thtee moments.So in fixe moments from
A to H: and in 8 moments from A to K. And
the differeoces of thofc moments, one and
three, and of thefe 6 and 8 are equal], that is
to fay two.
So alio of thofe quantities A C, and A E,and
of thefe, A H, and A K, the- diftetences C E,
and H Kate equall, and therefore differing
equally,as before.
%. Vepnlti- A Line is/aid to decreafe proportionally into a
on. Jhorter, when the poynt describing the fame in *~
quail times, cutteth ojfpgrts continually of the
fame prof onion to the lines from which they are
cut of.
£
-t—
Al| Mi
ft chaIenlP;ol*d H *« Auib«r,&
•mPThvldet<hoisv ^n(laior*
pub-
■zrfontat £
i ntutttn
2. 3 4 f 6-7 l^ioitnrz t?c*
-x
—«^
iJ)le bycducA/,
Printedft^c ft H o l m Oku,
•*•
Thb First Bookb.
Chap. T.
Of the Definitions.
Line isfaid to increafe equally, j.VefimtV
when thepoynt defcribing tbefame9 on
goeth ferward equall (paces
equalltimesfr moments.
in
#\\\ViiU\
Let Abe a poynt, from which a line is to be
jrawne by the motion of another poynt,
*hSw1n*e firft roo»ent,letB moue frcm
For examples fake. Lcr the line of the whole
fine 4 Z be to bee diminilhcd
proportionally ; let thepoynt diniinifhing the fame by his
motion
Cham. The fir $ B0*ke. 3
motion be b: and let the pioportion of each
part to the line from w€hit is cut off,bc as Q_
K to Q^S* Therefore in what proportion Q^
§ is cut in R, in the fame proportion(by thb
xo of the 6 of Euclid) Let 4 Z be cut in c. and
lb let b. running from J to $ in the firftmo-
mentjcutoff a c from a Z, the line or fine
^ Z remaining*
* And from this c Z let b proceeding in the
fecond moment,cut off the like fegment, or
part,asQ^R.to QS:and let that bee* d\
leauing rhe fine. dZ. From which therefore
in the third moment,let b in like manner,cqc
off the fegment d e, the fine e Z being left
bchiadc. From which like wife in the fourth
moment,by the motion of by let the fegmene
cfbe cut off, leauing the fine fZ. From
ih'isfZ in the fifth moment, let b in the fame
proportion cut off the fegmene fg> leauing
the fine g Z,-and fo forth infinkly. I fay thcr-
fore out ofthe former definition, that here
the line of the whole fine aZ9 doth
proportionally decreafe into the figne g Z, or into
any other laft iinc/tn which b ftaycth, and Co
in others.
Hence itfoUoweth tbatby this decreafe in /- A
quail moments (or times) there muft needes'Ulfo
bee left proportionall lines of the fame
Proportion.

Chapter 4 CORNERSTONES OF MATHEMATICS
Logarithm is the name chosen by Napier from the Greek words
logos, meaning "ratio", and arithmos, "number", and may be
translated as "ratio number".
The relation shows that the logarithm of 107 is zero, and the
logarithm of 107 (1 - 10" 7) = 0.999 999 9 is -1, which in many
ways is highly inconvenient for general calculation.
By dividing Naperian logarithms and their corresponding
numbers (antilogarithms) by 107, we nearly obtain a system of
logarithms to the base 1/e, because
10
n
1-
107
lim
n —» °°
1-
n
1
e
BRIGGS Henry
(1561-1630)
VLACQ Adriaan
(c. 1600 - after 1655)
DECKER Ezechiel de
KEPLER Johannes
(1571 -1630)
Thus, the often-repeated statement that Naperian logarithms
are "natural" logarithms - that is, logarithms to the base e - is
not true. It must be remembered, however, that Napier had no
thought of a base for his system of logarithms but arrived at it
from a comparison of uniform motion and retarded motion.
Napier published his results in 1614 under the title Mirifici
logarithmorum canonis descriptio ("Description of the
Wonderful Law of Logarithms"); an English translation appeared
in 1616 (see previous page). They were the outcome of about 20
years of calculations, for he had communicated his intentions
in 1594 in a letter to Tycho Brahe, who told Johannes Kepler.
In the summer of 1615, and again in 1616, Napier received
visits from Henry Briggs, then professor of geometry at
Gresham College in London. On a suggestion by Napier, it
was agreed that a change to a decimal base would be an
improvement, and a first installment of logarithms for the
numbers 1 to 1000, inclusive, to base 10 was published in 1617 in
London with the title Logarithmorum chilias prima.
Briggs's Arithmetica logarithmica (1624) contained
Briggsian, or "common", logarithms for the numbers 1 to
20 000 and 90 000 to 100 000, inclusive, the gap of 70 000
logarithms being bridged in 1627 by the two Dutchmen Adriaan
Vlacq and Ezechiel de Decker with Het tweede deel van de
Nieuwe telkonst, issued in English the following year as
Arithmetica logarithmica.
Johannes Kepler also prepared extensive logarithm tables,
used for accurate calculations of the revolutions of the planets
around the Sun. These tables appear in Kepler's Tabulae
Rudolphinae, printed in Germany in 1627; they are named in
honor of Kepler's patron, Emperor Rudolph II of Austria and
King of Bohemia.
Napier's description of the development of his ideas of his
logarithms, left in manuscript form, was published in 1619 as
Mirifici logarithmorum canonis constructio.

Section 4.5 Logarithms
153
As often happens when the time is "ripe" for a new discovery or
invention, Napier was not alone in developing improved
methods of converting operations of multiplication into
addition.
BURGI Joost(Jobst) The Swiss watchmaker Joost Burgi, maker of astronomical
(1552-1632) instruments and an indefatigable computer and assistant to
Johannes Kepler, Imperial astronomer in Prague, also
conceived a system of logarithms to facilitate the
multiplication of large numbers.
Where Napier decided on powers of (1 - 107), Burgi made the
better choice, (1 + 10~4); this made his power indices increase
as the power numbers increased, while Napier's decreased.
There was another difference between the work of the two men:
where Napier multiplied his powers by 107, Burgi chose 108; he
also multiplied his logarithms by 10 in his tables.
N = 108(1 + 10-4)L.
Burgi called 10 L the "red number" corresponding to the "black
number" N. If all black numbers are divided by 108 and all
red numbers by 104, we obtain what is nearly a system of
logarithms to the base e, because
/ l \io4 (
1 + comes near 1 i m 1 +
^ 104J n -> - ^
it agrees to four significant figures.
Burgi published his system in 1620 as Arithmetische und geo-
metrische Progress-Tabulen, effectively a table of antiloga-
rithms; it may be that he had begun his work in 1588, or even
1584. Napier is reported to have discussed his own results with
Brahe in 1594.
The matter of who was "first" must remain unsettled, but the
official priority belongs to Napier because of his publishing
date, 1614, six years before Burgi.

154
Chapter 4 CORNERSTONES OF MATHEMATICS
Logarithm, Exponent
Base
Antilogarithm
Definitions
We have
x = a log«* ,
that is, a logarithm is an exponent which defines to what power
the base a must be raised in order to give jc, called the anti-
logarithm.
Only positive real numbers a > 1 are acceptable as bases for
workable systems of logarithms. With bases a < 1, the higher
the antilogarithm, the smaller will be the logarithm - as
Napier came to realize. The number 1 is unthinkable as
a base, because logarithms will be undefined or, for 1,
indeterminate.
With a base a > 1
log x > 0,
log x < 0,
log 1 = 0
if x > 1
if0<x<l
The following practicable systems of logarithms are used:
lg x The common logarithm of x (to base 10)
(also log*, especially in older texts and on
electronic calculators)
In x The natural logarithm of x (to base e)
lb x The binary logarithm of x (to base 2)
loga x The logarithm of x to the base a
(also alog jc, especially in older texts)
We note the following equivalent notations:
loga2x = (logax)2
logalogax = log a (log a x)
Calculation Rules
Logarithms being exponents, calculation rules for logarithms
to all bases are the same as those of exponents:
a
m + n
_am.an
m — n
a
(am)n
n,
= am /an
= amn
m/n
a'
logxy =
logx/y
log xP
log V*
log x + log y ; x > 0, y > 0
= log*-logy ; x>0, y >0
; p log x ; x > 0
1 ,
= — log x : x > 0, p * 0
p

Section 4.5 Logarithms 155
Common Logarithms
Logarithms to the decimal base 10 - symbol lg or log - are
known as common logarithms or Briggs's logarithms.
A common logarithm may have the form
lg3300 = 3.5185...
Ig0.033 = 0.5185...-2 = -1.4815....
Mantissa where the decimal fraction is the mantissa and the integer
Characteristic part is the characteristic of the logarithm. Both were terms
suggested by Henry Briggs in his Arithmetica logarithmica
(1624).
Mantissa is a late Latin word of Etruscan origin, meaning
"addition" or "makeweight" - that is, something added to
make up the weight; it later came to acquire also the meaning
of "appendix".
The two logarithms quoted above demonstrate between them
the unique feature of the mantissa: it is linked exclusively to
the sequence, and order, of the digits of the antilogarithm,
irrespective of the position of the decimal point, if any.
Similarly, the characteristic has nothing whatever to do with
the sequence of digits, but depends exclusively on where the
decimal point lies.
The explanation is simple. We write:
3300 = 1000 x 3.3 0.033 = 3.3 x 0.01
lg 3300 = lg 1000 + lg 3.3 lg 0.033 = lg 3.3 + lg 0.01
= 3 + 0.5185... = 0.5185... - 2
= 3.5185... ( = - 1.4815... )
For logarithms ofN... the characteristic is
N > 1 one unit less than the number
of integer units of N
N < 1 negative, one unit greater
than the number of leading
decimal zeros of N
The method of writing negative logarithms with a separate,
negative characteristic facilitates calculation; electronic
calculators may use either form of representation.
Natural Logarithms
Natural logarithms - symbol In - are logarithms to the base e,
defined as
limits: p. 355 et seq. e= lim (l+ —
n ->.oo V Tl
In honor of Napier, they are also (wrongly) called Napierian
or Napier's logarithms.

1 56 Chapter 4 CORNERSTONES OF MATHEMATICS
The exponential function e* and logarithms to the base e
appear extensively in mathematics and physics. Since e* is its
own derivative, the laws of growth and decay, absorption in
optics, acoustics, and radioactivity, vibration and oscillation
phenomena in dynamics and electricity, etc., are most easily
expressed and worked with in terms of e x and In. Otherwise,
there are no advantages to the specific use of natural
logarithms; as an instance, logarithms to the base 2 are
customary in reference to radioactivity.
Logarithms of Negative and Imaginary Numbers
In Chapter 24, we shall define e z when z is a complex number,
and we shall then see that
ein = -1,
so we may take
In (- 1) = in.
As the imaginary unit i equals V-l , and In V-l = In (-1)3^2
= (1/2) In (-1), it follows that we may write
, . in
lni =— .
In fact, all numbers - real, imaginary, and complex - except 0
have logarithms.
p. 792
Changing the Base
It is sometimes desirable to change the base of a logarithm.
We have, by definition,
a loe« m = m .
The logarithm to the base b of both sides is
loga m • logfc a = logb m ,
from which
logbm
logam =
log b a
Example:
To find the natural logarithm of x, if its common logarithm is
2.77815 ..., we have
log a m = In x ;
logbm = \gx = 2.77815...;
a = e;
b = 10;
logb a = lge,
and
2.778 15
lnx & —I .
lg e
Using an electronic calculator, we find lg e & 0.434 29, and
2.778 15
0.434 29
^ 6.397 0

157
4.6 Mathematical Proof
Greek, axioma, "worth", "quality"
Latin, postulatum, "a thing
demanded"
Greek, hypo, "under"; thesis,
"a thing laid down"
Greek, theorema, "a subject for
contemplation"
Greek, lemma, "a thing taken"
Statements that are accepted without discussion or proof are
known as axioms or postulates.
In mathematics and other fields of logical reasoning, axioms
are used as a basis for the formulation of statements called
premises or hypotheses, which may result in propositions
called theorems. A lemma is an ancillary theorem whose
result is not the target for the proof.
PEANO Giuseppe
(1858 -1932)
The Peano Axioms
The natural numbers were axiomatically defined in 1899 by
the Italian mathematician and logician Giuseppe Peano in his
Aritmetices principia, nova methodo exposita. Later, Peano
modified his axioms to include also zero:
1. Zero is a number.
2. Every natural number or zero, a, has an immediate
successor a + 1.
3. Zero is not the successor of a natural number.
4. No two numbers have the same immediate successor.
5. The axiom of induction: Any property that belongs to zero,
and also to the immediate successor of any natural number
to which it belongs, belongs to all natural numbers.
Peano's first statement, after the axioms, is the most basic and
important numerical sentence of mathematics: 1 + 1 = 2.
\ {T-. j^ .J> J- jP. -T; -^ .■7^.-.>1"*. .-^ * • •■-. rt ,t^ j^ ._■& J- jP. -•?■ -^ .■7y.-.>1"*. .-^-.- .-^ -^...-^ /-^-.
£V y.i ■-■
•J
>
':>
1 + 1 = 2. One of Nicaragua's series of 10 stamps honoring
the world's most celebrated mathematical formulas.
■i- •</ -j? vj '-.■
<}.
\
i*
i
v.
i\

158 Chapter 4 CORNERSTONES OF MATHEMATICS
Proof by Deduction
A well-argued discussion of hypotheses based on established
postulates will lead to a conclusion whose validity depends
exclusively on the validity of the premises and on the
correctness of the reasoning, which then constitutes a proof by
deduction of the thesis put forward.
With deductive reasoning, a conclusion is reached by
working from established facts to particulars; that is, what applies
for, say, all polynomials in algebra will apply for any and
every particular polynomial, and in geometry, what applies
for all triangles will equally be true for any and every
particular triangle.
Proof by Induction
With the inductive method of proof, known from earliest times
but first described in 1838 by the British logician and mathe-
de MORGAN Augustus matician Augustus de Morgan, we base our reasoning on
(1806-1871) particular cases in order to arrive at a general conclusion.
For an inductive proof to be valid, that is, to be universally
true, it must be complete, which is to say that what is known to
apply for a particular case must be shown to apply for any and
every particular case of the given kind.
Proof by mathematical induction, or proof from n to (n + 1), is
not a method to discover new formulas or new truths but rather
one of reasoning, from an assumption, to results that may
confidently be expected to hold true.
To prove a theorem concerning a natural number n by
mathematical induction, we must
o establish that the theorem is true for some starting value N;
o assume that it is valid for a certain value n = p > N (the
induction hypothesis); and
o prove that it then also holds for the next higher value of n.
The theorem is then true for all n>N.
• Prove by induction that, for all natural numbers n,
n
V • n(n + l)
/Mi - 1 + 2 + 3 + ... + n = .
i = l
It is true for n = 1, so we start with N=l:
1 (1 + 1)
2
Let us now assume that the statement is true for an arbitrary
number n = p > N = 1 :
jr»=l + 2 + 3+...+p --^^^-
i = 1

Section 4.6 Mathematical Proof
159
QED
Under the induction hypothesis, the theorem must then also be
true for n = p + 1 = q:
P + l
5^i =1 + 2 + 3+. ..+p + (p + l) = P ^+ 1} + (p + 1)
j = l
p(p + l) 2 (p + l) (p + l)(p + 2) q(q + l)
"2 + 2" 2 "2
We know that the formula is true for n = 1; choosing p = 1, it
will be true also for /1 = 1 + 1 = 2, /i = 2 + 1, e£c; that is, the
formula is true for all integers n>l = N. QED
QED is the abbreviation of Quod erat demonstrandum, Latin
for "which was to be demonstrated". Euclid, in the 3rd century
B.C., used the Greek equivalent: c67iep >£8ei Sei^oci
Proof by Reductio ad Absurdum
Indirect Proof
EVCLEIDIS (c. 300 B.C.)
PYTHAGORAS (c. 500 B.C.)
Indirect proof, or proof by reductio ad absurdum, establishes
the truth of a statement by showing that the contradiction of it is
false and that, therefore, the statement must be true.
Let us illustrate by quoting two theorems dating back to the
days of Euclid and Pythagoras.
Hypothesis: The number of primes is finite.
Assume that p is the highest prime number in existence. Let P
be the finite product of all prime numbers,
P = 2 x 3 x 5 x 7 x 11 ... xp .
Now consider
Q = P+1,
which evidently cannot be composite as it would then be
divisible by one or more of the prime numbers making up the
product P. This is impossible, because every such division
leaves the remainder 1.
Q must then be a prime number greater than p or divisible by a
prime number greater than p. This contradicts the original
assumption, which must, therefore, be false.
Hence, the hypothesis is false; the number of primes is
infinite.
p. 78
Hypothesis: The square root of 2 is rational.
If the hypothesis were true, then V~2 = p/q, where p and q are
integers, and 2 = p2/q2 so
2q2=p2.
By the fundamental theorem of arithmetic the integers on the
left and right hand sides have unique and identical prime
factorizations. But, on the left side 2q2 must have an odd
number of factors of 2 since q2 must have an even number; on
the right side p2 must have an even number of factors of 2.

160
Chapter 4 CORNERSTONES OF MATHEMATICS
This is a contradiction.
Hence the hypothesis, which is that V~2 is rational, must be
false. So, y2 is rational. •
If uncomfortable with indirect proofs, one is in good company
with many mathematicians who take such proofs as a
challenge to find the positive version of them.
But couldn't medicine, for instance, also benefit from indirect
proof? If an investigation was designed to prove that a
hypothesis for cause or cure is false, one might be spared
"truths" that cause worrying, unwarranted hope, and waste of
time and money - the miracle of oat bran being just one,
curing everything from hemorrhoids and clogged arteries to
buck teeth.
Computer-Generated Proofs
p. 203
Modern proofs involve not only increasingly specialized and
complex mathematical methods but oftentimes also high-speed
computers, usually generating reams and reams of data. The
necessary hands-on verification of such proofs has become
cumbersome and time-consuming - sometimes close to
unmanageable - causing a dilemma of modern mathematics.
Theorems that long, in some cases hundreds of years, have
defied mathematicians have now been proved with the help of
the computer - notably the four-color map theorem.
The Incompleteness Theorem
GODEL Kurt
(1906-1976)
TARSKI Alfred
(1902 -1983)
There are many propositions in mathematics to which no
exceptions have been found and yet, no proofs have been
obtained to demonstrate whether they are true or not. As an
instance, all perfect numbers known today are even numbers;
we have no reason to believe that odd perfect numbers exist, but
no proof has been found to demonstrate their nonexistence.
Before 1931 it was believed that the axioms of arithmetic were
consistent and adequate to prove or disprove any
mathematical conjecture. However, in 1931 Kurt Godel, Czech-Austrian-
American mathematician and logician, published his famous
incompleteness, or undecidability, theorem stating that any
consistent formal system adequate to describe arithmetic must
contain statements which can neither be proved nor disproved
within this system. At the same time, Alfred Tar ski, Polish-
American mathematician and logician, studied the notion of
truth in formal systems.
Tarski's results together with those of Godel's show that there
is no systematic way to list the true statements in arithmetic.

161
4.7 Reliability of Digits and Calculations
Approximate Results and Errors
Teacher: John, zuhat is four phis five?
John: ...ohm...twelve.
Teacher: Wrong, you must stay after class and study!
John: Well, ma'am ... haven't you heard of tolerance for
some margin of error?
Accuracy
Absolute Error
Relative Error
An approximate value is not exact but might be accurate
enough for some specific consideration.
The accuracy is stated in the magnitude of the absolute or
relative error of the approximated value.
The absolute error is the difference between the approximate
value and the exact value.
The relative error is the absolute value of the quotient of the
absolute error divided by the exact value; it is often expressed
in percent (parts per hundred, %).
The exact values x\ = 2/3 and X2 = 2/15 are approximated to
a\ = 0.67 and a2 = 0.13, respectively. Determine the absolute,
relative, and percent relative errors of a\ and a2.
ei = o,\—x\ =
67
100
|ei/*l| =
1/300
2
" 3 =
1x3
3 x 67 - 100 x 2
3x100
= 0.005 = 0.5 %
1
300
e2 = a2-*2 =
2/3 2 x 300
13 2_ 15 x 13 - 100 x 2
" 15 '
I «2^2 I =
100
-(1/300)
15 x 100
300
2/15
1 x 15 1
= — = 0.025 = 2.5 %
Exact Approximate
value value
x1 = 2/3 ax = 0.67
x2 = 2/15 a2 = 0.13
2 x 300 40
Absolute
error
1/300
- 1/300
Relative
error
0.005
0.025
Relative
error, %
0.5%
2.5%
Although the absolute errors are equal, ax is a five times more
accurate approximation for xi than a2 is for x2 .

162 Chapter 4 CORNERSTONES OF MATHEMATICS
Reliable Digits
When a number is given in decimal notation, the absolute
error should not exceed a half unit of the last digit retained.
The approximate value 1.7 should not vary beyond ±0.05; the
approximate value 1.70 should not vary beyond ± 0.005 .
If the absolute error does not exceed a half unit in the last digit,
this digit is usually referred to as reliable. For instance, if
1299.6 (exact) is rounded to the closest 10 interval, 1300 is
obtained with absolute error
1300 - 1299.6 = 0.4,
indicating that all the digits of 1300 are reliable.
If, on the other hand, 1287 (exact) is rounded to the nearest
boundary of the 100 interval in which the number is situated,
we again obtain 1300, but this time the absolute error is
1300-1287 = 13,
showing that only the first two digits are reliable; the purpose
of the last two digits of 1300 is to define the magnitude of the
number.
If results are expressed in scientific notation,/? x 10", it is
generally understood that p represents digits that are reliable.
Thus, by expressing 1300 as 1.3 x 103 it is understood that only
1 and 3 are reliable digits in 1300, which can be stated to have
an accuracy of two significant digits.
Similarly, 1.30 x 103 indicates that 1300 has an accuracy of
three significant digits; 1.30 x 10~3 means 0.00130, that is, a
number with three significant digits.
Calculation with Approximate Values
When calculating with exact values, all significant digits can
be reported in the answer. For instance, when adding,
subtracting, or multiplying 5000 (exact) and 0.003 (exact),
it is correct to state the result as 5000.003, 4999.997, or 15,
respectively.
When calculating with approximate values or with blended
exact and approximate values, it is important to make sure
that the result does not include errors amplified by the
calculations.
Generally, a result should be rendered with no more
significant digits than there are significant digits in the value that
could carry the greatest absolute error.

Section 4.7 Reliability of Digits and Calculations
The three numbers 57.1, 3.304, and 34.16 are approximate
values with reliable digits, that is, with a maximum absolute
error corresponding to one-half of the last digit reported. 57.1
is evidently the term that could have the greatest absolute error,
± 0.05 - compared with ± 0.0005 for 3.304 and ± 0.005 for 34.16.
The sum should therefore be rounded to one decimal:
57.1
3.304
+ 34.16
94.564 => 94.6
Of the approximate values 1.005 and 0.37, the greater absolute
error could be carried by the term 0.37; their difference should
be given with two decimals:
1.005
-0.37
0.635 => 0.64
Of the approximate values 3.65 and 0.1020, the former could
present the greater absolute error; their product and quotient
should be rounded to three significant digits:
3.65x0.1020 = 0.3723 -> 0.372
3.65/0.1020 = 35.784... -> 35.8
The Case for Approximation
V2 is an exact notation, but if y2 or any other irrational
number is represented by nonperiodic infinite decimal
fractions it cannot be presented exactly as a finite decimal
expression. The greater the number of decimal digits given,
the closer the approximation will approach the true value, but it
will never be exact.
Periodic, or repeating, infinite decimal fractions also have an
interminable succession of decimals, and also represent
approximations of the true value; unlike nonperiodic
fractions they can, however, be written as common fractions, and
represent exact values.
To convert the infinite decimal fraction 1.166 666 ... to a
common fraction x, we write
10 x = 11.66
X = L1« 105 7
9x = 10.50 x = ~90~ = 6
For convenience, finite decimal fractions can be rendered as
approximations by limiting the sequence of decimals.

164 Chapter 4 CORNERSTONES OF MATHEMATICS
Measurements of physical quantities are generally subject to
accidental or systematic errors; these translate themselves
into inaccuracies of the results obtained, whose later digits
must be regarded with reserve, and may have to be removed by
further approximation.
We distinguish between two forms of approximation:
truncation and rounding.
Truncation
Three dots following a row of decimals indicate that further
digits have been suppressed. The number n, for instance, can
be written with nine correct decimals,
n = 3.141592 653...,
a degree of accuracy beyond the needs of practical calculation.
If only six - or four, or two - decimals are required, we obtain
by truncation,
7i = 3.141592... 7i = 3.1415... n = 3.14...,
or, replacing the equality sign = by the "approximately equal"
sign & :
n * 3.141592 n * 3.1415 n & 3.14
Valid Digits In a truncated expression, all digits are valid digits, that is,
they agree with those in the unabridged sequence, up to the
point of the cut. Thus, a truncated value will always be lower
than the exact value.
Rounding
Rounding of a number means replacing it by another number
having fewer significant decimal digits or, for integer
numbers, fewer value-carrying (non-zero) digits.
Thus, 123.7032 may be rounded successively to
123.703 123.70 123.7 124 120 100,
that is, to the nearest thousandth, hundredth, tenth, unit, ten,
and hundred, respectively.
Rounding may be carried out in two ways: by rounding down,
which is equivalent to truncation, and by rounding up the
last digit to be retained by one unit according to the rounding
rules of the International Standard Organization (ISO), ISO
standard 31/0.
The decision whether to round down or to round up depends on
the value of the leading digit of the sequence to be rounded off;
this is best illustrated by a practical example, using the real
number line.

Section 4.7 Reliability of Digits and Calculations
165
16.3 16.34
H h
co ^ w co t-
Tf Tf J Tf Tf
CO w " CO CO
co co ^o to co
16.35 16.4
Assume that we are to round the numbers
16.343, 16.344, 16.345, 16.346, 16.347
to the nearest hundredth: 16.34 or 16.35. We find immediately
that
o
o
o
16.343 and 16.344 shall be rounded down to 16.34;
16.346 and 16.347 shall be rounded up to 16.35; and
16.345 - halfway between 16.34 and 16.35 -
can be rounded either way.
Scientific Rounding
Shopkeepers' Rounding
Rounding Errors
If we decide to round all ...5 numbers consistently down or
consistently up, we introduce a systematic rounding error,
negative or positive.
To minimize the risk of systematic errors, ISO advocates the
use of scientific rounding, that is, all ...5 numbers are to be
rounded to the nearest even rounding boundary; thus,
95 05 15 25 35 45 55 65 75 85 95 05
—I—i—i—I—I—I—i—I—I—I—i—I—I—I—i—i—i—i—i—i—i—i—i—►
00
20
40
60
80
00
In commerce, so-called shopkeepers' rounding is the general
rule; all "5" pennies, cents, centavos, etc., are consistently
rounded up, which brings more coppers and nickels into the
till.
Error may be introduced if a number that has already been
approximated is again rounded. In the example above, 16.347
was - correctly - rounded up to 16.35; if rounded again, 16.35
becomes 16.4, which is evidently wrong, as 16.347 is nearer
16.3 than 16.4.
Thus, it is essential that all rounding processes be executed
in one step.
A way of guarding against mistakes of this kind is to place a
bar under or above a digit 5 that has resulted from a rounding-
up operation, e.g.,
26.146 & 26.15 = 26.15 & 26.1;
a 5 that remains unchanged after a rounding-down operation
may be marked by a dot above it, e.g.,
24.163 * 24.15 * 24.2.

1 66 Chapter 4 CORNERSTONES OF MATHEMATICS
Calculation with Boundary Values
Maximum accuracy in calculations with approximate values
is achieved when lower and upper boundaries are observed.
Assume the initial values
x=a±Aa; y=b±Ab.
Sum:
The lower boundary value is obtained by using the lower
values (a -A a) and (b - Ab) of the addends; the upper
boundary value by using the upper values (a + A a) and (b + A b):
[(a - Aa ) + (6- Ab)] < (x+y) < [(a + Aa) + (b + Ab)] .
Difference:
The lower boundary value is obtained by using the lower value
of the minuend (a - A a) and the upper value (b + A b) of the
subtrahend; the upper boundary value by using the upper value
(a + Aa) of the minuend and the lower value (b - Ab) of the
subtrahend:
[(a -Aa )-(6 +AM] < (x-y) < [(a + Aa ) - (b - Ab)]
Product:
The lower boundary value is obtained by using the lower
values (a - A a) and (b - Ab) of the factors; the upper boundary
value by using the upper values (a + A a) and (b + Ab) of the
factors, where the intervals contain only positive values:
(a -Aa)(b-Ab) < xy < (a + Aa) (b + Ab)
Quotient:
Again, assuming all the numbers are positive, the lower
boundary value is obtained by using the lower value (a - A a)
of the dividend and the upper value (b + A b) of the divisor; the
upper boundary value, by using the upper value (a + A a) of the
dividend and the lower value (b - A b) of the divisor:
(a -Aa)/(b + Ab) < xly < (a + Aa )1 (b - Ab)
Lower boundaries may only be rounded down; upper
boundaries only rounded up.
• The approximate values 4.376 and 2.356 are given, whose
digits are reliable, that is,
x = 4.3760 ± 0.0005
y = 2.3560 ± 0.0005 ,
4.3755 <x < 4.3765
2.3555 < y < 2.3565 ,
Find their sum, difference, product, and quotient with their
proper boundaries.

Section 4.7 Reliability of Digits and Calculations
1
Sum:
(4.3755 + 2.3555)
6.7310
6.731
Difference:
(4.3755 - 2.3565)
2.0190
2.019
Product:
(4.3755 x 2.3555)
10.30649 ...
10.306
Quotient:
(4.3755 / 2.3565)
1.856779 ...
1.8568
<
<
<
<
<
<
<
<
<
<
<
<
(x +y)
(x +y)
(x +y)
(x -y)
(x -y)
(x -y)
xy
xy
xy
x 1 y
x/y
x/y
<
<
<
<
<
<
<
<
<
<
<
<
(4.3765 + 2.3565)
6.7330
6.733
(4.3765 - 2.3555)
2.0210
2.021
(4.3765 x 2.3565)
i0.313 22...
10.313
(4.3765 / 2.3555)
1.857 991...
1.8580
!> Moo il«}ff
Ifie Boundary value for sheep
Counts onCy when grass isn't deep.
With head, more or (ess,
It can just be a guess,
founding up, rounding down as they teap.

168
Chapter 4 CORNERSTONES OF MATHEMATICS
4.8 Simple Calculating Devices
The Abacus
Counting on the fingers and writing in the sand at one's feet
were early forms of manual counting. The first device
actually invented to assist in arithmetic calculations was
probably the counting board, or abacus, which appeared
independently in various forms and in several parts of the ancient
world.
The early counting boards were tablets or trays of sunbaked
clay, or boards or blocks of wood, which were spread with a thin
layer of fine sand or dust in which the symbols were traced.
After a calculation had been completed and was no longer
needed, the designs could be easily effaced leaving the board
clean - tabula rasa — in readiness for the next operation.
Notable improvements of the counting board were the use of
more lasting materials, such as marble with the Greeks and
bronze with the Romans, and the provision of parallel grooves
in which markers in the form of pebbles or small balls could be
placed to serve as number tokens, symbolizing ones, tens,
hundreds, etc.
The Romans introduced a further improvement by providing
additional grooves between the existing ones, intended for the
"fives" of the Roman number system, making it ideally suited
to the Roman way of counting, which helped to maintain the
Roman numbers in operation for several hundred years -
well through the Middle Ages.
The origin of the word abacus can be traced to the Arabic
abq - meaning "dust" or "fine sand" - which became abax for
"sand tray" in Greek, and abacus in Latin. The Latin word
for "pebbles" is calculi (plural of calculus), a diminutive form
of calx, which means "stones".
To add 2769 (MMDCCLXVIIII) and 1987 (MDCCCCLXXXVII)
on the counting board, proceed in the following manner.
2769 1987
M ++ • 1000
D • • 500
C ++ ++++ 100
L • • 50
X • +++ 10
V • • 5
I ++++ ++ 1
[An abacus! Wow!
It can count!]

Section 4.8 Simple Calculating Devices
169
Begin by assembling all counters of the two numbers on the
left; this adds up to six I (1); two V (5); four X (10); two L (50);
six C (100); two D (500); and three M (1000).
Z 4756
M •••
D ••
L ••
X ••••
V ••
Counter arrangement,
intermediate
M ••••
D •
C ••
L •
X
V •
I •
Counter arrangement,
final
Abace, fad
[Get on with it!]
(Difficife eritfaciCe!
[Difficult gets easy!]
facitior quam faciCis!
[Easier than easy!]
%apide factum!
[Swiftly done!]
Suan-pan
Soroban
- Convert five I (1) to one V (5); carry it to the V groove,
making three V.
- One I (1) counter is left in the units groove.
- Convert two V to one X (10), carry it to the X line, making
five X counters.
- One V (five) remains in the "fives" groove.
- Carry the five X as one L (50) to the L groove, making three
L counters.
- No X (10) counter remains on the tens line.
- Convert two L (50) to one C (100), carry it to the C groove,
making seven C counters.
- One L (50) remains in the "fifties" groove.
- Convert five C (100) to one D (500), move it to the D line,
making three D counters.
- Two C (100) counters are left in the hundreds groove.
- Convert two D (500) to one M (1000) and carry it to the M
groove.
- One D (500) counter remains on the "five-hundreds" line.
- There are now four M (1000) on the thousands line.
The sum is MMMM D CC L VI = 4756 .
The form of abacus that we know today developed first in the
Far East - the suan-pan in China as early as the 11th century,
and the soroban in the 14th century in Japan. They consist
essentially of a wooden frame in which are mounted a number
of thin bamboo or metal rods, each with nine or ten colored
beads, corresponding to the counters of the counting board. As
many beads as there are units, tens, hundreds, etc., in the
numbers to be entered are collected in groups at the ends of
the rods.
The method of adding and subtracting with the abacus is the
same as for the counting board. Instructions how to use it for

170
Chapter 4 CORNERSTONES OF MATHEMATICS
multiplication and division, and how to extract square and
cube roots, can be found in old Chinese and Japanese works,
and in some early Western books, notably the 1610 edition of
Robert Recorde's Ground ofArtes.
The abacus used today in Far Eastern countries has a frame
divided into two sections, one with four "Earth" beads on each
rod and the other with a single "Heaven" bead, equivalent to
five beads. The Russian abacus, the s'choty, has ten beads on
each rod. The beads shown below may represent the whole
number 1 096 503 or a decimal fraction.
1 0
Modern soroban
9 6
0 3
Soroban o hajikimasho!
[Let's work the soroban!]
Gambarimasho!
[Give it a good try!]
23 x 613
10 9 6 5
Russian s'choty
The use of the abacus for multiplication and division is
described here for the Japanese soroban, which operates with
two kinds of counters in the form of balls or beads: Heaven
counters move down from above, and Earth counters come up
from below, to meet and be read at a "horizon" line.
Multiplication
Task: Multiply 23 x 613
Begin by placing the multiplier 23 at the left end of the soroban,
the multiplicand 613 to the right of it; and reserve space for a
five-digit product at the right end of the counting board.
Heaven
JUL
III ill
TfTTTTf
2 3
6 13
Earth
Multiply the units digit of the factor 23 by the units digit of
the factor 613: 3x3 = 9
Place counters to the value of nine units - one Heaven
counter plus four Earth counters - on the units line of the
product.
ii i ii
11 ±11 !±
T? ft? I
2 3
6 13

Section 4.8 Simple Calculating Devices
171
23x613
Yoku dekimashita.
[Good!]
Multiply the tens digit of the factor 23 with the units digit of
613: 2(0) x 3 = 6(0)
Add one Heaven and one Earth counter on the tens line of
the product; then, having completed operations with the
units position 3 of the multiplicand 613, remove that digit.
II
6 13 6 9
TT
2 3
11
TTJ
6 10 6 9
23x613 23x613
3.
2 3
Multiply the units and tens digits of 23 by the tens digit of
613: 3 x 1(0) = 3(0); 2(0) x 1(0) = 2(00)
Add three Earth counters on the product tens line, and two
on the hundreds line; then, having completed operations
with the tens position 1 of the multiplicand 613, remove
that digit.
ii ha
M
6 12 9 9
TfllTITff
2 3 6 0 2 9 9
±jL
&
23x613
f
Choi!
[Pay attention here!]
4. Multiply the units digit of 23 by the hundreds of 613:
3 x 6(00) = 18(00); 18 = 20 - 2
Adding 18 on the hundreds line is equivalent to adding two
counters on the thousands line and removing the two
counters on the hundreds line.
Tf TT II
2 3
6 2 0 9 9
23x613
Ji^-fte
Kantan desu ne?
[Isn't this easy?]
5.
Multiply the tens of 23 by the hundreds of 613:
2(0) x 6(00) = 12(000)
Operations completed with the hundreds position 6 of the
multiplicand 613, remove that digit. Add two counters on
the thousands line, and one on the ten-thousands line.
ii iiiii
Tf Tf II
2 3
14 0 9 9
6. The sought product is 14 099.
<fc<"ef SL& !
Yoku dekimashita!
[Yes, you did it!]

172
Chapter 4 CORNERSTONES OF MATHEMATICS
Warizan o yatte mimasho!
[Let's try division!]
Division
Task: 377/26
Place the divisor 26 to the left of the dividend 377. Beginning
with the units position of the 377, count off toward the left the
number of columns which are contained in 26; the second
column (A) to the left of those columns will be the units position
of the quotient.
ii
WtfTT
2 6
3 7 7
We shall use a method that follows ordinary longhand
division.
1. 26 goes into 37 once. Raise one Earth bead in the column to
the left of the units position of the quotient.
Lll
2 6 1 3 7 7
2. Multiplying the temporary quotient, 1, by the first digit of
the divisor 26 gives 1x2 = 2; return two Earth beads from
the hundreds position in 377.
TrnnTTT
2 6 1 17 7
3. Multiplying the 1 of the temporary quotient by the second
digit of 26 gives 1x6 = 6; return one Heaven and one Earth
bead from the tens position in the dividend 377.
Jozu desu ne.
[You are very skillful.]
TTmTTT
2 6 1 117
JJL1

Section 4.8 Simple Calculating Devices
173
4. As 26 does not go into 11, include one more digit to the
right; 26 goes four times into 117. Raise four Earth beads in
the column of the units position in the temporary quotient.
*>j6>9*u;fcjfc
Wakarimashita ka?
[Did you understand it?]
$9
Owari!
[Finished!]
Kotae wa onaji deshita ka?
[Did you get the same answer?]
6.
2 6 14 0 3 7
▲
5. Multiplying the 4 of the temporary quotient by the second
digit of 26 gives 4x6 = 24; return two Earth beads from the
tens position and four from the units position in 377.
TmiTTf
26 140130
As 26 does not go into 13, include a 0 immediately to the
right of the divisor 26; 26 goes five times into 130. Lower
one Heaven bead in the column immediately to the right of
the units position in the quotient.
LLJL
26 145130
▲
Multiplying the 5 of the quotient by the second digit of 26
gives 5 x 6 = 30; return three Earth beads from the tens
position in 377. Multiplying the 5 by the first digit of 26
gives 5 x 2 = 10; return one Earth bead from the hundreds
position in 377.
2 6 14 5 0 0
▲
8. The sought quotient is 14.5 .
A disadvantage common to all abaci is that a calculation can
never be checked except by doing it all over again, or by
keeping a record of it on paper. In old times, paper was
handmade from rags and consequently very expensive - hardly
within reach for most people. Also, although the Chinese had
known for about a thousand years how to make paper, it was
not introduced into Europe until the 12th century and did not
become common in the Western world until the 19th century.
Among those who master the abacus, its use is still often
preferred for basic operations - even over electronic
calculators. The abacus is very useful for teaching children the
fundamental operations of addition and subtraction, and the
meaning of place value in position arithmetic.

174
Chapter 4 CORNERSTONES OF MATHEMATICS
NAPIER John
(1550-1617)
Greek: rhabdos, "staff', "rod"
Napier's Bones
Many mechanical devices have been invented for
carrying out specifically multiplication, among them Napier's
bones, also called Napier's rods, described by John Napier,
Scottish mathematician and inventor of logarithms, in his
Rabdologiae in 1617.
RABDOLOGLfc,
SEV NVMERATIONIS
PER VIRGULAS
LIBRI DVO:
Cum Append ice dc expeditiC
fimo MVLTIPLICATIONIS
PROMPTTARlOr
Quibus acccflit & Arithmetic*
Localks Libei. vnts,
KrAttihort^j (jr Inventorcj I o an n a
Nbpbro, Baronet Mbr-
CHISTONII , &C.
StOTO,
HD 1KB V R G T,
Bxcudebtf vindtvu Hm> \6vj
Fol. i
RABDOLOGIiE
LIBER PRIMVS
De ufu VlRGVLARVM
nuracratricium in gcncrc.
C A P V T I.
De Fabric*, & infcriftione
Pir'itUrvm,
^57¾ Ab DO LOG! A til Art
** Computandi per Vir-
gulas numeratriccs.
VirgttU antem
numeratrices , funt vir.
guU quadrauy mobiles, ftmpltcium
notarum multiplu inferipu^ ad dif
ficiliores ^Arithmetics vulgaris ope*
rati ones facile & expedite perfcien-
das.
Virgularum itaques confiderabi-
mus Fabricam, & upim.
A F
Napier's rods constitute in essence a mechanical
multiplication table.
Multiplications were performed with the aid of rectangular
strips of bone, wood, metal, or some other suitable material. A
column of the multiples of the digit heading the rod was
inscribed or engraved on each rod, with a diagonal line
separating the tens - to be "carried" - and the units, as shown
in the drawing.

Section 4.8 Simple Calculating Devices
175
To illustrate the use of the rods, we choose the multiplication
of 1615 by 365 - the example described by Napier in
Rabdologiae. Place rods 1-6-1-5 beside each other as shown;
the results of the multiplication of 1615 by 3, 6, and 5 are found
to be 4845, 9690, and 8075, respectively - representative of the
partial products 484 500,96 900, and 8075, which are then added
in the usual manner.
1
A
/2
/3
/4
/5
/6
/7
/8
/9
6
/6
/1
/0
3//
4//
/1
4//
5/1
1
/\
/1
/3
/4
/°
/6
/7
/8
/y
5
/5
iy
/o
^
2/
/0
2//
3//
/^0
3//
/5
4//
X
3 • 1615 = 4845
5 • 1615 = 8075
6 • 1615 = 9690
1615
x 365
8075
9690
4845
589475

176
Chapter 4 CORNERSTONES OF MATHEMATICS
The Slide Rule
History
Mathematical computations can be performed with the help of
graduated scales which measure numbers as distances. In
their simplest forms, linear scales are used for direct addition
and subtraction of numbers. Linear scales have equal
distances between consecutive integer graduations.
111
23456789 10
n\ i 1 i I i It I i I i I i I i 1 i 1 i I i I i I i
II II I II
I I I I I I I I I
i^l i I i | iI i | iI i | iI i | i1 i | i I I | I I I | I
23456789 10
}
3.5 + 4.5 = 8
123456789 10
f 111111111111111111111111111111111111111111
i
1
1111 M ' I' I' |' 1' |' 1' |' I' |' 1' |' l' |' I' |jl' ["H1
123456789 10
9.5-5 = 4.5
Soon after Napier's publication of Mirifici in 1614,
logarithmic scales were beginning to be used for mechanical
computation. The drawing below shows a logarithmic scale to base
10 matched with a linear scale, which reads the logarithms of
the numbers on the logarithmic scale.
0
0.5
i 1
T
3 4
TT
6 8
10
1.5
i i i i i
20 30 40 60
100
lin
log
GUNTER Edmund
(1581-1626)
OUGHTRED William
(1574-1660)
DELAMAIN Richard
(? - c. 1645)
In 1620, Edmund Gunter, English mathematician and
designer/maker of navigation instruments, constructed a
"line of numbers" on which distances proportional to the
logarithms of numbers were marked off. Distances were
added and subtracted on the scale with the help of compasses, so
that multiplication and division of numbers could be
performed.
The English mathematician William Oughtred invented, in
1621, a rectilinear slide rule with two movable logarithmic
scales, and a circular slide rule, a device having two circular
logarithmic scales moving against each other to permit the
addition and subtraction of logarithms.
In 1630, Oughtred's former pupil Richard Delamain - a joiner
by trade but a capable mathematician, tutor to King Charles I -
published a 32-page pamphlet, Grammologia, or the Mathema-
ticall Ring, in which he presented the circular slide rule as his
own invention. He had sent his pamphlet to the king in 1629
for approval before publishing it.

Section 4.8 Simple Calculating Devices
177
William Oughtred then published his invention, in 1632, as
The Circle of Proportion and the Horizontal Instrument,
which started a lifelong quarrel between them. Delamain
countered, in the same year, with The Making, Description
and Use of a Small portable Instrument for the Pocket... Called
a Horizontal Quadrant. The war of words and pamphlets went
on, several people joining in, till the death of Delamain in the
civil war about 1645.
Oughtred's priority to the rectilinear slide rule has never been
in doubt, however, and his two logarithmic scales are the
undisputed precursors of the C and D scales of modern slide
rules.
The development of the slide rule gave it a few more
logarithmic scales, including a reciprocal scale, a cubic scale,
ROGET Peter Mark and trigonometric scales; in 1815, Peter Mark Roget - known
(1779-1869) from Rogefs Thesaurus - added so-called loglog scales for
calculations of and with exponential functions.
Experiments were made with better material - seasoned
wood from pear trees was found to be particularly suitable -
and the development of improved engraving machines raised
the accuracy of slide rules.
Many new scales were introduced during the decades
immediately following World War II, arranged on both
the front and the back of stock and slide in an ingenious
manner to maintain constant connection with other scales,
especially the basic C and D scales.
The advent of fast, low-cost, handheld electronic calculators
has largely forced the engineer's beloved slide rule to beat a
retreat into the science museums. But the satisfaction of
owning an electronic calculator and the fun of experimenting
with its buttons hardly measure up to the pride felt possessing
and mastering an advanced slide rule!
Berke Breathed; Washington Post
Writer's Group; © 1987

178
Chapter 4 CORNERSTONES OF MATHEMATICS
Construction
A slide rule consists of a stock, a slide (or slider), and a
transparent cursor (or runner) with a hairline to facilitate the
reading of the settings, or a system of hairlines, as the case
may be, for cooperation between the scales.
|iilllllll|llll|llll|illl|llll[llll|llll|lll^
i
K
£i.i I'l'iM'i'i'iMU'i'i'i'p'rri'i'i'i'i'i'i'i'i'i'iMM'i'i'i'i
|i|i|i|i|i|i|i|i|i|^iiii|iiiiyiii|iiiii . j . m Mmujl
I'l'I'I'I'I'I'I't'l'I'lji'lTI'lU'l'I'I'I'l
|t|l|t|l|t|t|l|'|MH|l ll|l||||llll| I I I
a *>• » , 2" . 3K 4 ? 6 f 8 I 10 \
a ' ^wf "I * IH^{UM' 1«hl'IMMM'1'l'>IMMHHnn 1 >nilnn}nniin|nn|iin nii|miWW*|f)+t
3 B o* 1 2 ' 3tt 4 5 6 J 8 A 'rt '
r
s
T
0 9<»5 0 91
■ 11 1111111 ■ 1 • ■»111.. • I i ( | | . I i |
0Q1
I i • i' i i it • i i i
|<| II Hill ll|tttt|tlti|ltltllttl|ttlthit|| I I II J I II I II III
•»6 7 J * »«-
-^
10
11
12
13
14
■» 10
't i A3,
•li'.».|.|ilil,M,l !.|/nl'I.I.I i '
*i r. tt 10 , 9 8 7 6 r> i
J* *' I | I | M |iiii|"|iIhiiIiii f •»• i I t • • • 'i i i li I i ill i i i I M l i I'll' I'M
0 c 09 1 it i? u i4 i5 9 2 I |rt|i '♦ I 5 6 7 8 9 10 it
° 09 | ,t ,7 13 14 15 t 2 : K ' *" " " *~ -
0Q5
i l 11111 il 11 ii'iiiiI ii| i i i
i lll|ll|l|llli|llll|
••6 7 8 9 K> «0 It I? II U 15 »70
|i]MM|illiltllltl|i|lil||tlllUIIII|lll|| I I I (MMI|||t||IUI||lll|||IUI|i |l|l||l|l||i|i|||||||
»»* »89 ' W wo n (2 J13 14 15 '2070
| °
11 'tin
i|ini|i|ii| i | I | i
'I'l'l'l'l'l'l'l'l'i'l i,l'l,ll|,l,|ll|iTill,l'|,|,l,lM'l,i'",ll',T1'l',l,'i
1-^..^,..,.1-1-1^1^-..,,,,.,,,,.,,,,,, , f «i Ti'i^oo
7 8 9100 1?n
iti|mi|nnlinilr|t|t|
3 4 5
3 4$
x*
6 7 8 9100 IPO
2 , 9 , 1 5 1.4 M I? 11 1.0»
. ? „ 1
I i I i t i I i iiui'iitiiiitii|H111n 1111111,, 1111,11,111111111 11.1it11.id *
x»
4 5 6
h' oi n n
\ , i i i . 1 , | i '.'. ' i .
7
0 7
8
Of,
I i 1111 ■ ■ >i|iiii| i 11 ilt i|il i t i|i| »' **
i,i|i,i,,,i|i|,,i|,|,|,|,|<|i|iH| I iii|iiii|llii|Mii|iMi|»iV'>q| . - -i-otxCroO
7i 30^ 55 40*0 5040 60*0 70 «090
.ii|»ii|Miir.i^iii|iiii,mi|iiii|Mi^i 11111. j ij\ \ 1111. yjj1111111^ „ „„0,r „.,
to point 2,
to point 3,
to point 5,
The stock and the slide carry several scales for a variety of
purposes:
C D are the basic scales - C on the slide and D on the
stock. They are graduated 0 - 10 in logarithmic
measure: if the length of the scales is 250 mm - a
normal value - the distance from point 1, the origin
of the scale, is,
250 x lg2 = 75.26 mm
250 x lg 3 = 119.28 mm
250 x lg5 = 174.74 mm.
A B These scales - A on the stock and B on the slide -
are also logarithmic, and graduated 1 - 100; this
means that every number on scale A, or B, is the
square of a corresponding number on scale D, or C,
permitting direct squaring of any number and
extraction of square roots.
These computations may, of course, be incorporated
into longer sequences of operations by developing
some ingenuity on the part of the user.
CI This standard scale, on the slide, is the reverse of the
C scale, running from right to left; the letter I stands
for inverse.
DI AI More sophisticated slide rules may also have one or
BI more reciprocal scales DI, AI, and BI.
K This scale, a fixed scale on the stock, is calibrated
logarithmically 1 - 1000, making it represent the
cubes of numbers on the D scale.
K'
Some slide rules may also have a moving cubic scale
K' on the slide.

Section 4.8 Simple Calculating Devices 179
S The S scale - S for sinus - has a double function.
Read from left to right, numbers in black represent
the sines of angles from 5° 8 to 90°; read right to left,
numbers in red represent cosines of angles from 0° to
84° 2 . Sines and cosines are read off the D scale.
P The P scale - where P stands for Pythagorean -
gives the cosine of an angle when the sine is known,
and the sine from a known cosine. It is used in
conjunction with the S scale to find better values of
sine and cosine.
T The T scale - T for tang ens - is used to find
tangents of angles; black numbers refer to angles
from 5° 8 to 45°, red numbers to angles 45° - 86°.
Tangents (and cotangents) are read on the D scale
and the CI scale.
ST Some slide rules have an ST scale, also known as an
arc scale. For very small angles, the sine and
tangent are very nearly equal, and equal to the arc of
the angle, in radian measure.
Cursor hairline on ST 2° 25 reads
sin = tan = arc = 0.0395.
Igx Most slide rules also have a fixed linear scale on the
stock, which corresponds with the D scale and gives
the mantissas of D numbers.
LL Most slide rules have at least one, generally three
LL - for loglog - or exponential scales, used for
computing powers with fractional exponents, solving
exponential equations, and calculating the values of
hyperbolic functions.
In their heyday, slide rules were designed for the most
diversified special uses - for mechanical strength
calculations of truss bridges, steel wire ropes, and reinforced concrete
structures; for electric engineers designing generators and
motors, and building power transmission lines; for architects
and surveyors; for economists and businesspeople; and even
simplified versions ... for schoolteachers and physicians.
Advanced Deluxe Slide Rules
There were also double-sided deluxe slide rules for those who
demanded the utmost in mathematical versatility. Besides all
the usual scales mentioned above, they had several of the
special scales described below.
CF DF One side of the stock and slide of the slide rule was
provided with F - for folded - scales; if C and D
were graduated in jc, scales CF and DF showed nxf
and went from n through 10 to 10 7C.
Besides providing scales for the handy calculation of
all relations involving n, the folded scales gave the

180 Chapter 4 CORNERSTONES OF MATHEMATICS
additional advantage that you never risked getting
"out of bounds" with multiplication.
CI DI on these slide rules were reciprocal scales graduated
in 1/x or 10/jc.
CIF is a reciprocal scale of 1/nx or 10/nx.
W 1 For increased accuracy, these scales have twice the
W1' length of the corresponding C and D scales by being
W2 divided into parts; Wl and Wl' are graduated
W2• in V*, W2 and W2" in Vlo7 .
LL The most advanced slide rules featured up to eight
exponential scales e* and e~x, covering powers from
0.000 01 to 0.999 1 on four scales and from 1.000 9 to
100 000 on another four scales.
Calculations
For maximum accuracy, multiplication and division made
with a slide rule should be carried out an scales C and D,
whenever possible. Operations 82 x 62 and 0.082 x 620 are
performed in the same way, and provide answers that are just
an array of digits; the place of a decimal, if any, must be
determined separately.
Let's dust off a slide rule and illustrate with a few problems.
Find product 2.1 x 3.45 .
- Place 1 of scale C over 2.1 on scale D.
- Move cursor hairline to C 3.45 .
- Read answer, approx. 7.25 on D scale.
t
3.45
D
2.1 7.25
The exact answer is 7.245 .
Find product 3.2 x 6.3 .
If C 1 is placed at D 3.2, the factor C 6.3 will extend beyond the
end of the D scale. To avoid this,
- Place cursor hairline on D 3.2 .
- Place CI 6.3 under hairline to divide by 1 / 6.3 .
- Read answer, approx. 202, on D below C 1.
A rough estimate gives 3 x 7 = 21 to place the decimal point:
20.2. The exact answer is 20.16 .

Section 4.8 Simple Calculating Devices
181
Find quotient 19.2189 .
- Place C 1 on D 1.92 .
- Move hairline to CI 8.9, to multiply by 1 / 89 .
- Read answer, approx. 216, on D under hairline.
A rough estimate is 20190 & 0.2. The answer is 0.2157.
Find quotient 7.1/3.4.
- Place cursor hairline on D 7.1 .
Place C 3.4 under hairline.
- Read answer, approx. 2.09, on D at C 1 .
The result is based on lg (7.113.4) = lg 7.1 - lg 3.45.
3.4
D
2.09
7.1
(The answer is 2.088... .)
Find product 312 x 27 x 45 x 175 x 26.
- Place hairline on D 312 .
- Place CI 27 under hairline.
- Move hairline to C 45 .
- Place CI 175 under hairline.
- Move hairline to C 26 .
- Read answer, approx. 172 on D under hairline.
The answer is approximately 1.72 • 109 .
Find
15 x 36 x 37
23x29
Place hairline on D 15 .
Place C 23 under hairline.
Move hairline to C 36 .
Place C 29 under hairline.
Move hairline to C 37 .
Read answer 29.9 on D
under hairline.
Find 19 • V35 .
- Place hairline on A 35 .
- Place CI 19 under hairline.
- Read answer, approx.
112.4, on D scale.
Find
24x87x37
34x11
7TT • 77 • 87 .
Rearrange factors to
37 24
34 ' 11
- Place hairline on D 37 .
- Place C 34 under hairline.
- Move hairline to C 24.
- Place C 11 under hairline.
- Move hairline to C 1.
- Place CI 87 under hairline.
- Read answer, approx 206.6, on
D at C 1.
Find 28 • V240 .
- Place hairline on K 240 .
- Place CI 28 under
hairline.
- Read answer, approx. 174.0,
on D scale.

182
Chapter 4 CORNERSTONES OF MATHEMATICS
The Quipu
Quipu The quipu is generally associated with the Incas, who ruled
Peru before the Spanish conquest. Similar devices were
used by several other Indian tribes and are also described
in Chinese and Persian documents from the 6th and 5th
centuries B.C.
The Inca quipu consists of a number of color-coded cords.
Knots of various kinds are tied on these cords to represent a
variety of information. The Incas had no written language;
instead, the administration of their mighty empire depended
heavily on the use of the quipu.
An important use of the quipu was to record numbers, as in
trade, keeping accounts, and for calendars, but the knotted
strings might also have served as mnemonics for the
recording of important historical events, astronomical data,
mythology, and legends.
There is evidence to support the theory that the knots and the
cords of the old Inca quipu were in a decimal system of
numeration: A knot tied farthest away from the cord
designated units, a nearer knot specified the next position in the
system of numeration, etc.; absence of a knot symbolized
zero; larger knots were multiples of smaller ones.
Still, today, devices related to the Inca quipu are used by
shepherds in the Andes for keeping an account of their herds.

183
Chapter
5
COMBINATORICS
Page
5.0 Historical Notes 184
5.1 Multiplication Principle 186
5.2 Permutations 188
5.3 Combinations 196
5.4 Samples with Replacement 199
5.5 Graph Theory 201
5.6 Magic Squares and Their Kin 205

184 Chapter 5 COMBINATORICS
5.0 Historical Notes
Combinatorics is the name of a branch of mathematics that is
concerned with the selection of objects - called elements -
from a given set of elements.
Combinatorics traces its history back to ancient times when it
was often closely associated with number mysticism, as in the
Chinese book I Ching from 2200 B.C.
In the Western world, interest in combinatorial mathematics
was awakened in the 17th and 18th centuries and largely
stimulated by questions being raised about odds in gambling
and other games of chance, which led to the creation and
development of probability theory.
The Italian astronomer, mathematician, and physicist
Galileo Galilei studied the relative probability of various sums
of points occurring when rolling dice. Other important
contributors in the field of probability theory were the
Frenchmen Pierre de Fermat and Blaise Pascal, the German
von Leibniz, and the Swiss brothers Nicolaus and Daniel
Bernoulli.
In 1736, the Swiss mathematician Leonhard Euler solved the
celebrated problem of the Konigsberg bridges - the question
whether it would be possible to make a tour of the city and
return to the starting point by crossing all of its seven bridges
just once; this feat marks the beginning of graph theory.
Combinatorial geometry includes problems of covering,
packing, and symmetry. A celebrated conjecture in packing
theory, posed in 1611 by Johannes Kepler, is that the most
compact way to stack spheres is into four-sided pyramids.
This method has been used for the display of fruit long before
Kepler's time, but the mathematical proof is as yet elusive.
Combinatorial methods are widely used in most branches of
mathematics and in many areas of applied mathematics; they
are of special interest in probability and statistics, and in
computer science.
As with most branches of mathematics, the exact scope of
combinatorics cannot be defined; only in textbooks does it
exist as an independent subject.
Saint Ttttfs Qatne
In a well-renowned pub in Scotland a party had assembled
consisting of fifteen whisky drinkers and fifteen temperance
apostles - emissaries of the Quild of Whiskey (Distillers and the
League of Teetotalitarians who were arguing volubly about the
excellence or perniciousness, as the case might be, of the pub's
drinl^ables - to which, understandably, the publican too/^ e?(ception.
GALILEI Galileo
(1564-1642)
de FERMAT Pierre
(1601-1665)
PASCAL Blaise
(1623 -1662)
von LEIBNIZ
Gottfried Wilhelm
(1646 -1716)
BERNOULLI Nicolaus
(1695-1726)
BERNOULLI Daniel
(1700 -1782)
EULER Leonhard
(1707 -1783)
KEPLER Johannes
(1571 -1630)
Kepler's Conjecture

Section 5.0 Historical Notes
185
So he ordered everybody on the premises to form a line, counted off
nine several times, booting every uniner" out before resuming his
counting ... until there were finally only fifteen emissaries left and
- to and behold! - they zuere aliivhisl^ey drinkers. - Uisghe Beata!
Now, how did the publican manage this little piece of fiddling?
Our story is a variation of the "not so chancy" Ludus Sancti
Petri, or Saint Peter's Game, known from medieval times -
typically a ship was struck by a heavy storm and, to lighten
and possibly save the vessel and some of her passengers, half
the passengers were thrown overboard. In Europe the fifteen
favored persons were most often Christians, the other fifteen
Jews. St. Peter, who usually presided, arranged the
passengers in such a way that the Christians were saved.
In a Jewish version, Saint Peter was replaced by the Spanish
Jew Ibrahim ben Meir Ezra - a foremost mathematician,
astronomer, and translator of science from Arabic to Hebrew
- who saves his fifteen pupils, the other fifteen passengers
being drowned. An Arab version favors the Muslims and
throws the infidels overboard; an East Indian story lets fifteen
good men be saved and fifteen thieves be drowned. A Japanese
land-based version describes a stepmother who wishes to
deprive her stepchildren of their inheritance.
Beginning with four favored (f) and five unfavored (u), the
complete arrangement is
4f; 5u; 2f; 1 u; 3f; 1 u; If; 2 u; 2 f; 3 u; If; 2 u; 2f; lu.
Speakers of English could memorize this arrangement - if
able to spell - from the order of vowels in a mnemonic rhyme,
where a = 1, e = 2, i = 3, o = 4, and u = 5,
Jrom numbers' aid and art,
9{ever will fame depart.

186
Chapter 5 COMBINATORICS
5.1 Multiplication Principle
k«.W.~.-.
sugar cone
waffle cone
sugar cone
waffle cone
sugar cone
waffle cone
vanilla
chocolate
strawberr
praline
pear
vanilla
chocolate
strawberry
praline
pear
vanilla
chocolate
strawberry
praline
pear
vanilla
chocolate
strawberry
praline
pear
vanilla
chocolate
strawberry
praline
pear
vanilla
chocolate
strawberry
praline
pear
Tree Diagram (Product Tree,
Factorial Tree)
The following selections of ice cream cones are available:
A: 1, 2, or 3 scoops, all the same flavor
B: Sugar cone or waffle cone
C: Vanilla, chocolate, strawberry, praline, or pear flavor
The tree diagram (product tree, or factorial tree) illustrates the
number of possibilities.
As the illustration shows, if no flavors are mixed the total
number of varieties available is
3 • 2 • 5 = 30 .
This is an example of the multiplication principle:
If k different selections can be made in succession, and
the first selection can be made in n\ ways, the second
selection in n<i ways, the third selection in 713 ways, etc.,
then the total number of selections is
/i l -/12 -/13
-n>k
* * *
ttandshafgs; Tennis Matches
Si?cty-four tennis players meet at a welcome party before the start
of a tournament. They greet each other with handshafgs; how
many? !And how many matches will be played to find a winner?
Answer: svipvmi £9 !styvysjmvy sjqz

Section 5.1 Multiplication Principle
187
If the order of selection is noted, as in the example below, the
collection is called an ordered sample.
Appetizers
• Potato Skins
• Escargot
• French Onion Soup
• Shrimp Cocktail
• Oysters on the
Half Shell
• Marinated
Mushrooms
'Dinner Menu
Entrees
• Broiled Orange
Roughy
• Chicken Kiev
• New York Strip Steak
• Breaded Jumbo
Shrimp
• Filet Mignon
• Crab Louie
• Fillet of Sole
• Fettuccine Alfredo
Desserts
• Black Forest Torte
• Chocolate Chambard
Cake
• Cherry Cheesecake
• Deep Dish Apple Pie
• Cheese and Fruit
Plate
• Vanilla Ice Cream
• Peppermint Sherbet
When the waiter asks if you are ready to order, you answer
that you want a three-course meal but want to study all the
possible selections before you order.
How many three-course meals can you compose from the
menu?
6 appetizers • 8 entrees • 7 desserts
= 336 three-course dinner selections.
(The waiter won't like you very much.)
In days past, a physician made house calls in four villages
A,B, C, andD.
Find in how many ways the physician can travel
a: between A and D, and
b: from A to D and back to A
a:
4-2-5 = 40 different ways.
b:
4-2-5.5.2-4 = 1600 different ways.

188
Chapter 5 COMBINATORICS
5.2 Permutations
Drawing by Cheney; © 1989
The New Yorker Magazine, Inc.
Latin, per, "thoroughly"
mutare, "to change"
Left to right, top row: Arthur Tylor Winston, Tylor Winston
Arthur, Winston Arthur Tylor. Bottom row: Tylor Arthur
Winston, Winston Tylor Arthur, Arthur Winston Tylor.
Permutations are ordered arrangements of a finite number of
elements, either of all of the available n elements or of a part p
of them. Two permutations that contain exactly the same
elements but not in the same order are regarded as different.
The notation
p
n* p
Size
Tuple
Mapping
Map
Functions: Chapter 10
reads, "The number of permutations of n elements, selected p
at a time" or "The number of ordered p -tuples that can be
formed from n elements".
Equivalent notations are nPp , Pp , and P(n, p).
The number of elements (p) used at a time is the size of the
tuples (permutation).
Braces are used for unordered sets, parentheses for ordered
tuples, thus: {A, B, C} = {B, C, A}; (A, B, C) * (B, C, A).
There are six possible permutations of the three letters of the
triple {A, B, C}:
(A,B,C); (A,C,B); (B,A,C); (B,C,A); (C,A,B); (C, B, A); 3^3 = 6,
that is, the number of permutations of all the elements of the
collection of 3 elements equals 6.
And here are the six ordered pairs taken from {A, B, C}:
(A,B) (A,C) (B,A) (B,C) (C, A) (C, B); 3^2 = 6.
We think of the permutation of (A, B, C) to (B, A, C) as an
action, function, or mapping that takes a triple and transposes
its first two elements. This allows us to compose several
permutations by following one action by another and gives a
dynamic element to the study.
The map that sends A to B and B to A, and leaves C fixed, may
be displayed: A
X
B
C
B

Section 5.2 Permutations
189
All Elements Distinct
A<B"D
b<a-d
B^D-A
D<A~B
The first element of a permutation of n elements can be chosen
in n different ways, the second element in (n - 1) ways, the
third in (n - 2) ways, etc., until all n positions have been
filled. Thus, the number of permutations that are possible with
n elements is the product of all natural numbers from 1 to n,
inclusive, that is, n factorial,
nPn = 7i(7i-l)(7i-2)...3-2-l = n\
In order to find all possible permutations of a given set of
elements, proceed systematically: for numerical elements in
their natural order, by magnitude; for letter arrangements, in
lexicographical order.
The tree diagram on the left shows that {A, B, C, D} has 24
permutations if all four letters are selected in every instance.
The diagram illustrates that
4P4 = 4x3x2x1= 4! = 24.
• The word NUMBERS has seven letters, all different. When
all letters are selected in every instance, we have
7P7 = 7! (= 1x2x3x4x5x6x7) = 5040 permutations.
• If 12 gates are available for 12 racehorses, the number of ways
the gates can be assigned is
12 ! = 479 001600 .
• The 52 cards of a conventional deck of playing cards can be
arranged in
52-51-50...3-2-1 = 52!
= 80 658175 170 943 878 571660 636 856 403 766 975 289 505 440 883 277 824 000 000 000 000
different ways.
Permuting the cards at the improbable rate of one permutation
a second round the clock, year in and year out, would require
2.5 • 1060 years to run through the entire lot. This is more than
1050 times the age of the Universe. •
Sometimes a limited number of elements are to be selected
from the original collection. The tree diagram on the left
demonstrates the 12 permutations of the collection
{A, B, C, D} when only two letters are selected:
4P2 =4x3 = 12
The general formula for the number of permutations of a
collection when all elements are different is
B
4
°<
A
B
C
p
n (n - 1)... [n - (p - 1)] =
n\
(n-p)\
where n andp are positive integers with n>p.

Chapter 5 COMBINATORICS
For the above (A, B, C, D), when only two letters were selected,
we have
_ 4 ! 24 no
4P2 = 43 = (4_2)!=T=12,
which shows that the formula works.
Why? Let n elements be available forp positions.
At the second selection, n - 1 elements remain to be placed in
p - 1 positions.
For the second element, n - 1 selections can be made, for the
third Ti-2 selections, etc.; for the last element the number of
selections is n - (p - 1).
We have
nPp = 7i(7i-l)(7i-2) ...[n-(p - 1)] ,
which may be rearranged to
nPp = n (n - 1) (n - 2) ... (n-p + 1).
Rewriting,
_ 71(71 - 1) (71 - 2) ... (71 -p + 1) [(71 -p)\]
nFP " (n-p)\
gives
which is the general formula for the number of permutations
of a collection when all elements are different.
• When in NUMBERS only two letters are selected at a time,
the number of permutations is
7 »
7P2 = (7 - 2) ! =42*
• In how many ways can the starting order be posted in an eight-
member relay cross-country skiing team if
a: all 8 members will take part in a race;
b: only 4 members - chosen from the 8 members - will
take part?
a:
8 ! = 40 320 different ways.
b:
8 !
(8 - 4) !
= 1680 different ways.
A group of Australian businessmen, traveling by air, plans
a visit to eight states in the continental U.S.

Section 5.2 Permutations 191
In how many ways can the order vary if
a: the states may be visited in any order;
b: the group decides to visit New York State first;
c: the group decides to start in New York State
and finish in California?
a:
8 ! = 40 320 permutations.
Thus, the order of visiting the eight states may vary in 40 320
different ways.
b:
Since the first state has already been chosen, we are left with
the number of permutations of seven states.
7! = 5040 different ways.
c:
Since the first as well as the last state to be visited have already
been chosen, we are left with the number of permutations of six
states, all six taken at a time:
6 ! = 720 different ways.
In how many ways can seven people be placed
a: at a bar counter;
b: at a round table? ykili&v
J%v>Xv>>>y^X*X"X*«%^^
OeX'X'X'X^X'X^X'X'X'X^
cX*XwX,XvXvX#X*X*X,il
fcX^XwXvX^'X'X'X'X^X'J
WBW
:88:½¾^^
ooooooo
a:
7! = 5040 different ways.
b:
Rings Making a ring instead of a row is equivalent to starting out
with one element whose position has already been determined.
6 ! = 720 different ways.
Inversions
If two elements in a permutation of distinct elements are in
reverse order relative to their normal or natural order, they
constitute an inversion.
Even A permutation is said to be even if it contains an even number
Odd of inversions; it is odd if the number of inversions is odd.
The number of transpositions that are required to return a
sequence of elements to their natural order is even or odd
according to the number of inversions in the arrangement.

192 Chapter 5 COMBINATORICS
In the permutation (cedb a),
element c precedes b and a => 2 inversions
e precedes d, 6, and a => 3 inversions
d precedes b and a => 2 inversions
b precedes a => 1 inversion,
making eight inversions in all; hence, (c e d b a) is an even
permutation.
A transposition interchanges two elements in a permutation.
A transposition always alters the number of inversions by an
odd number: if c and a in (c e d b a) are interchanged, we have
a new permutation (ae d b c), where
element e precedes d, 6, and c => 3 inversions
d precedes frandc => 2 inversions,
making five inversions, three less than before.
Transposing two adjacent elements alters the number of
inversions by one unit. Changing reverse-order elements d b
to natural-order b d permutes (ae db c)to(ae b dc), where
element e precedes b, d, and c => 3 inversions
d precedes c => 1 inversion,
making four inversions, one less than the former five
inversions.
If, instead, we reverse the normal-order ae to e a, in the new
permutation (e a db c),
element e precedes a, d, 6, and c => 4 inversions
d precedes b and c => 2 inversions,
making a total of sijc inversions, one more than the former
five inversions.
Cyclic Permutations
The shifting of an entire ordered sequence of elements one or
more steps forward or backward - the first element taking the
position of the last, or vice versa - without changing the order
of the elements in the sequence is a cyclic permutation:
(ab c de) =$ (b c de a) => (c de ab) =$ (de ab c) => (e ab c d) => (ab c de)
The number of elements in the collection being permuted is the
Degree degree of the permutation.
A cyclic permutation of degree two - (a b) => (b a) - is a
Transposition transposition.

Section 5.2 Permutations
193
The Fifteen Puzzle
This popular puzzle was invented in the latter part of the 19th
century. Initially in random order and with the empty space
in the lower right-hand corner, the fifteen counters are to be
rearranged in numerical order.
There are more than 20 • 1012 possible permutations but only
half of them admit a solution: the puzzle can be solved only if
the total number of inversions in a given display is even.
This is because each move is a transposition of the blank (we
may think of it as 16) with some other square.
1 2
IV
!•:
5
1 15
BffllS
1
5
10
13
3
6
11
14
r ^8
4 |
7 I
V
12 1
■
Solvable
2 precedes 1 => 1 inversion
8 precedes 5, 6, 7 => 3 inversions
15 precedes 13, 14 => 2 inversions
Total: 6 inversions
Rubik's Cube
1 1
1 5
I 9
1 13
2
6
10
14
3 4 I
•I*
7
8 |
11 12 |
>■:
■
Solved
No inversions
mmm
8
I 12
^¾¾¾¾¾
•!»?»!»!»!»!»!'?»SSf»f»?>!'!»S!'!»!»!>!'!»!*!>S!>!*SS!»?»?»,^
*y M '"^
«
10
11
15
13
^*:'W7:,jA|'A'jA''.'.*jA*i
14
:¾¾¾¾¾¾¾¾¾¾¾¾¾^^
Unsolvable
4 precedes 3 => 1 inversion
8 precedes 5, 6, 7 => 3 inversions
15 precedes 12,13,14 =>3 inversions
Total: 7 inversions
Of course, a recent craze - enlightenment or frustration - is
Rubik's cube, consisting of 26 colored, rotatable cubelets (no
cubelet at the center) whose faces are of different colors, yet
every cubelet is similar, giving more than 1019 possible
arrangements (permutations). The challenge is to arrange
the cubelets to form a cube with every face displaying only one
of the six colors.
Identical Elements
The number of permutations is reduced when a collection
contains identical elements. The number of permutations of n
objects, q of one type, r of another, s of another, etc. to a total ofp
objects, is given by
n
r\ s\
Find the number of distinct permutations of the letters in
a: MISSISSIPPI b: MATHEMATICS c: COMBINATORICS
when all the original letters are used once each.

Chapter 5 COMBINATORICS
a: In MISSISSIPPI the letter I occurs 4 times, S 4 times, P twice,
M once.
11! 11!
(11-11) ! 4! 4! 2 ! 1 ! 0 ! 4 ! 4 ! 2 ! 1 !
= 34 650 permutations.
b: For MATHEMATICS,
111
(11 - 11)! 2! 2! 2! 1! 1! 1 ! 1 ! 1 !
11!
0 ! 2 ! 2 ! 2 ! 1 !
c: For COMBINATORICS,
13!
= 4 989 600 permutations.
0 ! 2 ! 2 ! 2 ! 1 !
= 778 377 600 permutations.
On analysis, the active part of an insulin molecule is shown to
consist of 30 amino-acid links, bound together in a chain-like
manner with two free ends. The number of different amino-
acid links is:
3 Glycine 1 Serine 1 Glutamine 3 Phenylalanine
1 Alanine 2 Threonine 1 Lysine 2 Tyrosine
3 Valine 1 Aspartic acid 1 Arginine 2 Histidine
4 Leucine 2 Glutamic acid 2 Cysteine 1 Proline
Find the number of permutations for linking the amino acids
together.
30_!
(30 - 30)! 3 ! 1! 3 ! 4 ! 1 ! 2 ! 1 ! 2 ! 1! 1! 1! 2 ! 3 ! 2 ! 2 ! 1!
* 1.598 987 629 x 10 27 permutations.
• In the Welsh island Anglesey, a few miles west of Bangor,
there is a village with the quaint name
Llanfairpwllgwyngyllgogerychwyrndrobwll-llantysiliogogogoch,
which means, in English,
The Church of St. Mary in the hollow of white hazel
near the rapid whirlpool, and the Church of St. Tysilio
of the red cave.
The fifty-eight letters of the name include eleven Ts, seven g's,
six o's, five y's, four each of n, r, and w, three each of a and i,
two each of c and h, and one each of b, d, e, f, p, s, t -
** 6.793 215 • 1055
(1 !)7 (2 !)2 (3 !)2 (4 !)3 5 ! 6 ! 7 ! 11 !
- clearly a name that will appeal to people with acute
permutationitis.

Section 5.2 Permutations
195
Latin Squares
Standard Form
Diagonal Latin Square
Mutually Orthogonal
Euler Square
BOSE Raj Chandra
(b. 1901)
SHRIRKHANDE S. S.
PARKER Ernest Tilden
(b. 1926)
A Latin square of the n-th order is a permutation of n symbols
arranged in n rows and n columns; it is in standard form if
the first row and first column consist of the natural (original)
order:
A
B
C
D
B
A
D
C
C
D
A
B
D
C
B
A
If the diagonals also have a permutation of the n symbols, it is
called a diagonal Latin square:
A B C D
C D A B
D C B A
B A D C
If all pairs of symbols are different when two Latin squares of
the same order are superimposed, the Latin squares are
mutually orthogonal and the square formed is known as an
Euler square:
A
B
C
D
B
A
D
C
C
D
A
B
D
C
B
A
a
c
■
+
d
b
b
d
c
a
c
a
b
d
d
b
a
c
Aa
Be
Cd
Db
Bb
Ad
Dc
Ca
Cc
Da
Ab
Bd
Dd
Cb
Ba
Ac
In 1959 Bose, Shrirkhande, and Parker, working in the U.S.,
demonstrated that Euler squares exist for any order, except two
and six.
Latin squares are of importance in the design of experiments
that will be subjected to statistical analysis. If four species of
peas A, 5, C, D are to be tested with four separate fertilizers a, 6,
c, d, the plots can be laid out in the combined way shown, so
each cultivar and each fertilizer appear in each row and
column.
* * *
Home Cookgd Combinatorics
8¾¾¾^
Si few minutes before their train was due to leave, siT^ people
ordered si?t eggs, over easy, one minute frying time each side. The
coo (^ at the Home Cooked Combinatorics Cafe, a great fan of
Martin Qardner, had one frying pan, with room only for four eggs.
She needed only three minutes to carry out the order; how?
Beginning first minute: She fried eggs 1, 2, 3, and 4 on one side;
End first, beginning second minute: She turned eggs 1 and 2 over, put 3 and 4
aside, and placed 5 and 6 into the pan;
End second, beginning third minute: Eggs 1 and 2 done, removed; 3 and 4
now returned to the pan, sunny side down; 5 and 6 were turned over;
End third minute: Eggs 3, 4, 5, and 6 done.

196 Chapter 5 COMBINATORICS
5.3 Combinations
Combinations are concerned only with the selection of objects
from a collection and, unlike permutations, disregard order.
The current mathematical terminology is unfortunate and
confusing, since in everyday language the word combination
often conveys a sense of order. For instance, the combination
of a combination lock is, in mathematical terms, not a
combination but simply a specified order of digits.
We know that the six possible permutations of two letters from
the sequence A, B, C are
(A,B) (A,C) (B,A) (B,C) (C,A) (C, B).
The above (A, B) and (B, A) are made up from the same
combination, the unordered set {A, B}. Likewise, (A, C),
(C, A) both arise from {A, C}, and (B, C), (C, B) from {B, C}
Consequently, the three possible combinations of two letters
from the sequence A, B, C are:
(A, B) (A, C) (B, C).
The notations
G)
both denote the number of combinations, and read
"The number of combinations of n elements selected p at a
time" or, in short, "n choose p"; or, in the case of the latter
notation, simply "n over pn.
Equivalent notations are nCp,Cp , andC(n,p).
Because ( n ) is the number of ways we can choose p jc's from
Binomial Coefficient p. 140 x/?/
the factors (1 + x)11, it is the binomial coefficient for xP in this
expansion.
Size The number (p) of elements used at a time is the size of the
combination.
We found that the collection {A, B, C, D} has 24 permutations
when all 4 elements are selected at a time. Since the order is
disregarded in combinations, the collection {A, B, C, D} has
only one combination when all 4 elements are selected at a
time. If, on the other hand, only 3 elements are selected at a
time, there are 4 combinations:
{A,B,C} {A,B,D} {A, C,D} {B,C,D}
Since each combination of a collection has the potential of
forming permutations, there is a simple relation between
combinations and permutations.
C -
n P 9 \p.

Section 5.3 Combinations
197
In the collection {A, B, C, D}, the 4 combinations
{A, B, C} {A, B, D} {A, C, D} {B, C, D}
would each form 3! = 6 ordered triples, giving a total of 4 x 6
= 24 ordered triples formed from {A, B, C, D}. We obtain the
relation
p
n* p = \n^p) p\ or rv^p — (
Each of the nCp combinations may be ordered in p ! ways to
make p-tuples, and each of these p-tuples can be completed in
(n - p) ! ways to make a permutation of n. So,
and we now have the general formula for the number of
combinations of a collection of different elements :
n ! fn \ n !
nCP = (n-p) ! p ! ; \p/ = (n-p)\ p !'
where n andp are positive integers with n>p.
We have
9 !
\3/ " (7-3) ! 3 ! " ™> \0/ "
(7 - 3) ! 3 ! ' \0/ (9 _ o) ! 0 !
= 1
What is the total number of matches in a round-robin tennis
tournament with thirteen contestants?
Every one of the 13 contestants meets 12 other players and plays
a total of 12 matches. Each meeting is mutual between 2
players, so the total number is
13-12 ne% ^
—-— = 78 matches
or, using the formula,
13 ! 13 • 12
(!)-
(13-2) ! 2 !
= 78.
Find the number of combinations in the word NUMBERS,
selecting at a time
a: 2 letters; b: 6 letters.
a:
7 !
7C2 = -77:—o\ \ 0 \ = 21 combinations.
b:
7 !
iCa = 77;—n. , n , = 7 combinations.
(7 - b) ! b !
A committee-crazed organization has 18 members; every
committee is composed of 4 members, and no same 4 members
may form more than one committee.
Find the maximum number of committees.
18 !
= 3060 committees.
(18-4) ! 4 !

198
Chapter 5 COMBINATORICS
12 men and 14 women apply for a total of 8 assignments in
a research project that will take place in a secluded
underground site, placing extreme demands on congeniality and
cooperation between all participants.
Find the number of possible combinations of the work crew
a: there is no stipulated men-women quota;
b: as many men as women should be employed.
a:
All applicants may be considered indistinct.
(12 + 14) !
-r 7 = 1562 275 combinations.
[(12 + 14)-8] ! 8 !
b:
Select the women and men separately, then apply the
multiplication principle.
f 14 ! \f
^(14-4) ! 4 1^(12-4) ! 4 !
12!
= 495 495 combinations.
^m> gzH
18 mice were placed in two experimental groups and one
control group, with all groups equally large. In how many
ways can the animals be placed into the three groups?
There are 18 animals to select from for the first group, 12 for
the second, and 6 for the third. To find the total number of
combinations, the multiplication principle is applied.
f 1QI \ f 101 \ f
18!
12!
^(18-6)! 6!A(12-6)! 6lJ\(6-6) ! 6!
6!
= 17153136 combinations.
Systems of Blocks of Elements
KIRKMAN Thomas Penyngton
(1806 -1895)
The digits of 123456789 can be arranged in blocks of triples so
that each pair of digits appears only once:
{1, 2, 3} {1, 4, 7} {1, 5, 9} {1, 6, 8}
{4, 5, 6} {2, 5, 8} {2, 6, 7} {2, 4, 9}
{7, 8, 9} {3, 6, 9} {3, 4, 8} {3, 5, 7}
Amateur mathematician and English clergyman Thomas
Penyngton Kirkman showed in 1847 that the requirement for n
elements to be arranged in a system of blocks of triples where
each pair of elements appears once - no more or less - is that
the number n divided by 6 gives a remainder of 3 or 1. In our
example, we have 9 elements: 9 divide4 by 6 gives the
remainder 3.
Since the 1950s much research has been done in this area of
combinatorics, resulting in criteria to form other systems of
blocks of elements. Applications exist in data processing and
telecommunications.

199
Samples with Replacement
In sampling with replacement, we imagine a bag containing
n objects. A sample of size p is made by picking an item from
the bag, recording it, and tossing it back into the bag, and so on
until an ordered p-tuple of values has been recorded.
Every sample may be selected as many times as there are
objects in the collection. A sample is replaced by a like sample
before the next sample is selected.
The number of samples formed with n objects, p at a time,
with replacement, is
nP.
Find the number of three-letter formations of the word CUP,
employing
a: samples with replacement;
b: permutations;
c: combinations.
a: 33 = 27 samples with i
(C, C, C) (C, C, U)
(U, U, U) (U, U, C)
(P, P, P) (P, P, C)
(C, U, P) (P, U, C)
(P, C, U) (P, C, P)
(C, U, C) (P, C, U)
b: 3 ! = 6 permutations.
c: 1 combination.
replacement
(C, U, U)
(U, C, C)
(P, c, C)
(U, P, C)
(P, U, P)
; these are
(C, C, P)
(U, U, P)
(P, P, U)
(C, P, U)
(U, P, U)
(C, P, P)
(U, P, P)
(P, U, U)
(U, C, P)
(U, C, U)
The code of a combination lock is selected from six of the
digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, where digits may be repeated; for
instance,
0
9
0
2
2
7
The number of possible lock combinations is
106 = 1000 000.
Using the digits
a: 1,3, 5, 7, 9 b: 0,2,4,6,8
find the maximum number of positive integer numbers
composed of one, two, three, four, or five digits that can be formed
when the same digit may be repeated 5 times in each number.
a: 5 + 5x5 + 5x52 + 5x53 + 5x54 = 3905 integer numbers.
b: A zero leading an integer number has no place value, so
4 + 4x5 + 4x52 + 4x53 + 4x54 = 3124 integer numbers.

200 Chapter 5 COMBINATORICS
• A licensing agency uses a system of any two letters from
the English alphabet, followed by three digits (1 through 9),
followed by any one letter from the English alphabet; for
instance,
EM 374 W
The greatest possible number of licenses that can be issued
using this system is
(262)(93)(261) = 12 812 904.
• The five lamps of an optical paging system can be switched to
show a steady light, a flashing light, or no light. When all
lights are off, nobody is being paged.
IE]
How many people can be individually paged by the system?
There are 3 choices (steady, flashing, and off) for the five
lamps; thus,
35 possibilities.
When no light is on, nobody is being paged; thus,
35 - 1 = 242 individuals can be separately paged.
• Braille is the language of the blind, who read, with their
fingertips, characters composed of raised dots in the paper,
arranged in a two-by-three pattern.
o • • o • o • o
• o • o • o o •
o o • o • o oo
I L L E
This system is an example of permutations, with repetition, of
two elements - dot and no-dot,
26 = 64.
Deducting one unit for the "six-no-dot" configuration, we are
left with 63 characters, which are enough to represent the letters
of conventional alphabets plus ten digits and the necessary
punctuation marks and diacritic signs.
• o
• o
o o
B
R
o o
A

201
5.5 Graph Theory
The Konigsberg Bridges
KANT Immanuel
(1724 -1804)
HILBERT David
(1862 -1943)
The former German city of Konigsberg, now Russian
Kaliningrad, on the banks of the Pregel and on two islands in
the river, is famous for its bridges and for two of its sons:
Immanuel Kant and David Hilbert.
Kant, the great philosopher, based his entire thinking on non-
negotiable mathematical decrees; although sometimes placed
among mathematicians, Kant did not advance mathematics.
Hilbert was a mathematical universalist, one of the leading
mathematicians of the 20th century.
Of old, Konigsberg had seven bridges connecting the several
parts of the town. This started a debate whether it would be
possible to make a complete tour of the town and return to the
starting point by crossing all of the bridges just once.
tffr" i\ \ \L « mjV-T" H
Map of Konigsberg,
circa 1740
Location of the bridges

202
Chapter 5 COMBINATORICS
EULER Leonhard
(1707-1783)
Graph
Network
Vertex
Arc
Valence
CAYLEY Arthur
(1821 -1895)
The problem was settled in 1736 by the Swiss mathematician
Leonhard Euler, who demonstrated the impossibility of the
task - at least one bridge would have to be crossed twice.
Euler reduced the city plan to what is today known as a graph
or network, in which land areas are represented by dots -
called vertices (singular: a vertex) - and bridges are shown
as lines between the vertices, called arcs. The valence of a
vertex is the number of arcs that originate or end at the vertex.
Euler formulated the following general law for the solvability
of the problem:
A path traversing the network by crossing every
segment just once is possible only if the network is
connected and has at most two vertices with odd valence.
Networks of KOnigsberg and Kaliningrad
7 bridges 8 bridges
B B
The Konigsberg network had four vertices - A B C D - all of
odd valence: A with five; B, C, D with three segments each.
Consequently, the suggested tour is impossible.
An eighth bridge was built between B and C in 1875, directly
connecting the banks of the river. Of the four vertices, A still
has five segment connections, and D has three; but B and C
have four; that is, only two vertices have an odd number
of segments, and the suggested tour of the city is possible
- starting at one of the vertices with an odd number of
connections and ending at the other. If we build a ninth bridge
between A and Z), we can also meet the requirement that the
tour should begin and end in the same place.
Which of the figures below can be drawn without lifting the
pen and without repeating lines? Give an Euler argument.
A.
B.
•aouajBA ppo t#ia\ saopjaA g /L\uo iajqissoj
•aouajBA ppo i#ia\ saoi^iaA (g <) f fajqissod ^0^
'80U8p3A U8A8 UB S13T{ X8^I8A iCl8A8 ^jqiSSOj
How Many Possible Connections?
D
a
v
Digressing from the Euler paths, we ask how many ways exist
to connect a specified number of dots, representing, e.g., cities
that need road and telephone connections. Arthur Cayley
showed that the number of connections of n dots is nn ~ 2 . To
connect five dots, there are 55 " 2 = 125 possibilities; to connect
eight, 262 144 possibilities.

Section 5.5 Graph Theory
203
Euler Paths vs. Hamilton Paths
Euler path
Hamilton path
HAMILTON William Rowan
(1805 -1865)
Irish mathematician
An Euler path traverses every segment of a network once, with
no restrictions as to the number of times each vertex may be
passed through; a Hamilton path goes through every vertex
once only, with no obligation to traverse all segments but
ending at the starting point. They are important in traffic
planning, electric circuit design, and many other
applications of networks when the task is to find the shortest path for
the least expensive installation.
There is no simple necessary and sufficient condition for the
existence of a Hamilton path.
The Four-Color Map Theorem
Chromatic Number
GUTHRIE Francis
(1831 -1899)
If a map can be drawn as a continuous closed curve (or broken
line) on a sphere or a plane - returning to the starting point - it
requires only two colors; if it does not return to the starting
point, it requires three colors.
The question arises: What is the minimum number of colors
needed for any map on a sphere or a plane? The least number
of colors that is sufficient so that regions with common
boundary-line segments on a surface are distinguished by
different colors is known as the chromatic number of that
surface.
In October 1852, Francis Guthrie - then a young mathematics
student, later to become professor of mathematics at the
University of Cape Town - found that he could color a map
using just four colors so that no two neighboring countries
would have the same color.
••••••••••••••••••••
• ••••••••••• T? ••••••
• •••••••••• • r•••••••
•••••••••••••••••••
• • • • •
• • • • • 4
• • • • •
• ••••,
• • • • *y
• • • *^
t t t
Guthrie conjectured that four colors would do, but he could not
present a satisfactory mathematical proof of this thesis.

Chapter 5 COMBINATORICS
de MORGAN Augustus
(1806 -1871)
CAYLEY Arthur
(1821 -1895)
KEMPE Alfred Bray
HEAWOOD P. J.
(1861 -1955)
pp. 448 - 50
YOUNGS J.W.T.
RINGEL Gerhard
(b. 1919)
Klein bottle: p. 380
APPEL Kenneth
(b. 1932)
HAKEN Wolfgang
(b. 1928)
Mobius strip: p. 380
FRANKLIN Philip
(1898 -1965)
Twelve-Color Map Problem
The problem was brought to the attention of the eminent
English mathematician Augustus de Morgan, who proved that,
irrespective of individual size and shape, five countries
cannot occupy such positions on a map that every one of them
will have a border common with all of the other four.
This suggests but does not prove that four colors might suffice
for the coloring of any map; the difficulty of the problem lies
in the fact that the proof must apply for all imaginable
configurations of any number of countries, of any size and shape.
Mentioning Guthrie as originator of the problem, de Morgan
submitted it to several well-known contemporary
mathematicians, but it failed to attract general interest until, in June
1878, Arthur Cayley brought it before the London Mathematical
Society. Many amateur and professional mathematicians
have since tried their luck at the problem.
In 1879 the British lawyer Alfred Bray Kempe claimed to have
proved the four-color map conjecture. Kempe substituted nodes
for countries and connected nodes of bordering countries
by arcs to reduce the problem to a set of "unavoidable cases".
For more than a decade Kempe's proof stood but, in 1890,
P. J. Heawood pointed out that Kempe had missed a particular
case. Heawood used Kempe's approach and Euler's
polyhedron formula (Chapter 12) to prove that five colors are
always enough for maps on the plane or sphere; moreover, he
gave a general formula for the chromatic number, computed to
be less than or equal to a number p. In 1968 J. W. T. Youngs
and Gerhard Ringel proved that p equals the chromatic
number, except for a Klein bottle.
The four-color problem was solved by an approach similar to
that of Kempe's in 1976 by Kenneth Appel and Wolfgang
Haken of the University of Illinois. Their proof was made
with the assistance of a computer, which allowed analysis of
a large collection of cases. After correcting the initial
argument, there was now - almost a hundred years after Cayley's
invigorating presentation of the problem - an infallible
mathematical proof of the four-color map conjecture.
Mathematicians still hope for an easily accessible proof
- obtained without the need for computer assistance - but no
such proof has yet been found, if in fact it is possible. The
present proof is available only to the most determined.
Thus, the plane and sphere have the chromatic number 4; it
has further been proved that the torus, the cylinder, and the
Mobius strip all have the chromatic number 7. These numbers
can all be computed by Heawood's formula, given in 1890. In
1934, Philip Franklin proved that the chromatic number for a
Klein bottle is 6. A condition for all chromatic numbers is that
a colony must not require the same color as its parent country.
In addition to the four-color map problem - now exalted to the
position of a theorem - there is a twelve-color map problem: If
on a plane or sphere each country has at the most one colony,
requiring the same color as its parent country, at most twelve
different colors are needed to distinguish the political regions
on a map (chromatic number: 12).

205
6 Magic Squares and Their Kin
A square with 9, 16, 25 ... n2 boxes, called cells, filled with
integer numbers - all different - is called a magic square if
the sums of the numbers in the horizontal rows, vertical
columns, and main diagonals are all equal.
The magic square is of Chinese origin, being first mentioned
in a manuscript from the time of Emperor Yu around 2200 B.C.
This square had 3x3 = 9 cells, each with Chinese characters
equivalent to 1 through 9, and giving the sum 15 in all
directions.
It was inevitable that a square with these "magical", and
therefore mysterious, qualities should appeal to astrologers
and cranks of all descriptions.
Thus, a square of one cell containing a digit 1 - exhibiting the
magic of producing the sum 1 in all directions - was
considered to represent the eternal perfection of God (= Number
One), as explained to the lay mind by one Cornelius Agrippa
(1486 - 1535), an astrologer by profession.
The unfortunate fact that a magic square with 2x2 cells cannot
be constructed was considered proof of the imperfection of the
four elements: air, earth, fire, and water; some self-styled
Great Thinkers attributed the failure to Original Sin.
If the integers in a magic square are the consecutive numbers
from 1 to n2, the square is said to be of the n-th order, and the
magic number is equal to
n (n2 + 1)
The simplest magic square possible is one of the 3rd order,
with 3x3 = 9 cells containing the first nine integers, 1 ... 9, and
with the magic sum 15 along eight lines: 3 horizontal,
3 vertical, and 2 diagonal.
Only one arrangement of digits, and its mirror image, is
possible for a 3rd-order square:
4
3
8
9
5
1
2
7
6
and mirror image
2
7
6
9
5
1
4
3
8
They may be rotated into four positions each, permitting eight
3rd-order magic squares in all.

206
Chapter 5 COMBINATORICS
DURER Albrecht
(1471 -1528)
The first magic square to appear in the Western world was,
in all probability, the one depicted in the upper right-hand
corner of a copperplate engraving called "Melencolia § F by
the German artist-mathematician Albrecht Diirer, who also
managed to enter the year of engraving - 1514 - in the two
middle cells of the bottom row.
16
5
9
4
3
10
6
15
2
11
7
14
13
8
12
1
The Diirer magic square is of the 4th order, that is, it has
4 x 4 = 16 cells containing the first sixteen natural numbers so
arranged that adding any one of four horizontal rows,
four vertical columns, and two main diagonals will produce
the magic sum,
4(16 + 1)
= 34,
as in all 4th-order magic squares.
16 + 3 + 2 + 13 = 34
5+10+11+8 = 34
9 + 6 + 7+12 =34
4+15 + 14+1 = 34
13+8+12+1 = 34
2 + 11 + 7 + 14 = 34
3 + 10 + 6 + 15 = 34
16+5 + 9+4 =34
4 + 6 + 11 + 13 = 34
16 + 10+7 + 1 = 34

Section 5.6 Magic Squares and Kin
207
Besides these sums, the Dxirer square yields the same magic
sum when adding other configurations of cells as shown in the
drawings below, where, for clarity, filled circles denote the
numbers to be added.
16 + 13+1 + 4 = 34
10 + 11+7 + 6 = 34
j
3 + 2 + 14 + 15 = 34
5 + 8+12+9 =34
2 + 8+15+9 =34
5 + 3 + 12 + 14 = 34
13+10+4+7 = 34
16 + 11+6 + 1 = 34
»
rr-— t
HZ
1XXX
t-% itzl Si
KA h?£ t±4
In addition to the magic squares already described, there are
also magic multiplication squares and magic division
squares. In the former, the products of all numbers in every
horizontal row, vertical column, and main diagonal are the
same; in the square illustrated here, the product is 216, e.g.,
12 • 1 ■ 18 = 216
18-4-3 = 216, etc.
If we reverse the order of the numbers in the main diagonals,
we obtain a magic "division square", in which the first figure
of every row, column, and main diagonal divided by the
quotient of the second and third figures will always give the
same result, and so will the last figure divided by the quotient
of the second and first figures, e.g.,
T = 6
18
J8 _^8 _
q — q d, etc.
3

Chapter 5 COMBINATORICS
CHENG TA-WEI
In 1593, the Chinese mathematician Cheng Ta-wei published a
book from which we have culled the following 6th-order magic
square:
FRANKLIN Benjamin
(1706-1790)
52
14
53
11
55
.9
50
16
61
3
60
6
58
8
63
1
4
62'
5'
59'
.7'
*
2
64
13
5*
12
4
54
10
56
15
49
fco
*6
fcl
43
41
18
48
29
*35
28
38
26
40
31
33
36
30
37
27
39
25
34
32
45
19
44
22
42
24
47
x17
Franklin's Magic Square
27
9
32
14
28
1
29
11
25
16
6
24
2
20
7
34
15
33
4
22
3
30
17
35
13
31
21
12
26
8
36
18
23
5
19
10
Every row, column, and main diagonal adds up to 111.
The fascination of magic squares does not stop with 6th-order
squares.
The American statesman and scientist-inventor Benjamin
Franklin constructed a magic square of the 8th order, whose
every horizontal row and vertical column add up to the magic
number 260 or, if you stop halfway, to 130 for its quarter-
squares.
But the magic does not end there: Every four-cell mini-square,
say,
3 62 6 59
60 5 or 58 7'
sums to 130; every four cells symmetrically placed in
relation to a mini-square, e.g.,
52 13
11 54
61 4
6 59
14 51
55 10 ;
and
53 12
9 56 '
60 5
8 57 '
11 54
55 10
add further to the magic by presenting the sum 130.
But there is more to come: Adding the numbers on the rising
and descending diagonal dotted lines will also give the
expected sum, 260 - all in all, a magnificent combinatorical
feat, worthy of the highest praise.
From the ordinary 8th-order magic square it is but a short step
to the refinement of letting the numbers follow the movements
of a chess knight across a chessboard - "up two and over one"
or "up one and over two".

Section 5.6 Magic Squares and Their Kin
209
EULER Leonhard
(1707-1783)
1
30
47
52
5
28
43
54
48
51
2
29
44
53
6
27
31
46
49
4
25
8
55
42
50
3
32
45
56
41
26
7
33
62
15
20
9
24
39
58
16
19
34
61
40
57
10
23
63
14
17
36
21
12
59
38
18
35
64
13
60
37
22
11
Beverley's Magic Square
5
54
15
64
17
34
27
44
14
63
6
25
56
43
18
35
53
4
55
16
33
26
45
28
62
13
24
7
42
57
36
19
3
52
41
58
23
8
29
46
12
61
10
49
32
39
20
37
51
2
59
40
9
22
47
30
60
11
50
1
48
31
38
21
Feisthamel's Magic Square
46
43
54
17
52
31
2
15
55
18
45
42
3
16
51
30
44
47
20
53
32
49
14
®
A
19
56
41
48
13
4
29
50
58
21
12
5
40
33
@>
27
9
6
57
24
61
28
39
36
22
59
8
11
34
37
26
63
7
10
23
60
25
62
35
38
Jaenisch's Magic Square I
50
23
10
61
48
59
6
3
11
62
49
22
7
4
47
58
24
51
@
9^
60
45
2
5
63
12
21
152
<D
8
57
46
14
25
40
33
20
53
44
31
37
34
13
28
41
32
19
56
26
15
36
39
54
17
30
43
35
38
27
16
29
42
55
18
Jaenisch's Magic Square II
The famous Swiss mathematician Leonhard Euler published a
paper in 1759, where he showed how to complete partial knight's
tours and to construct symmetric knight's tours. Euler's
chessboard magic squares inspired many amateur and
professional mathematicians to design chess-knight magic
squares.
First to design a magic knight's tour was William Beverley
(1814 - 1889), an artist and designer of theatrical effects; his
publication appeared in the Philosophical Magazine in 1848.
Like Franklin's square, the Beverley magic square shown
here adds up to 260 - 130 in its quarter-squares - for all
horizontal rows and vertical columns; also like Franklin's
square, all four-cell mini-squares sum to 130, but the
agreement ends there. The Beverley magic square can be
considered four 4th-order magic squares put together. The
more limited versatility of the Beverley magic square
compared with that of Franklin is a consequence of the
restriction imposed by the formal pattern of movement of the
chess knight, whereas Franklin was free to arrange his
numbers in the square largely at his own discretion.
The second to construct magic knight's tours was Carl
Wenzelides (1770 - 1852); his first one was published in the
German chess magazine Schachszeitung in 1849.
The magic square of Feisthamel fulfills the requirement
of adding up to 260 in all horizontal rows, vertical columns,
and, unlike Beverley's magic square, also in the two main
diagonal lines.
The chessboard square is characterized by a kind of axial
symmetry with reference to the vertical midline; similarly
placed cells on opposite sides always add up to 65.
The path of the knight across the chessboard is not continuous
but consists of two halves, 1 ... 32 and 33 ... 64, cells 32 and 33
being both on the 5th horizontal line from the top with no
possibility for the knight to bridge the gap except by riding
piggyback on a rook.
The (1) IN and (64) OUT cells are at either end of the 4th line
from the top of the board.
One of Jaenisch's magic squares (I in the margin) has an
uninterrupted numerical path 1 ... 64 for the knight with the
finishing cell 64 inside the board but so placed that the knight
can reach the starting point 1 from it for a new round, an
elegant feature. All horizontal and vertical lines of this
square add up to 260, but the diagonals do not, the bottom left to
top right diagonal giving 256 and the top left to bottom right
giving 264, which makes the square only semi-magic.
Another Jaenisch magic square shown here (II in the margin)
is also semi-magic; the horizontal and vertical sums are 260,
but the diagonals add up to 192 and 328, respectively. The path
of the knight is continuous 1 ... 64, with a regular jump back
to 1. The square exhibits a form of radial symmetry, in that
numbers contained in cells symmetrically placed relative to
the mid-point of the board always have a difference of 32 units.

210
Chapter 5 COMBINATORICS
Composite Magic Squares
Sometimes you can construct large magic squares by fitting
together magic squares of lesser orders.
As an illustration, consider the simple 3rd-order magic
square
(1)
2
7
6
9
5
1
4
3
8
magic sum = 15
and build it up into similar squares by adding, repeatedly, 9 to
every cell, obtaining successively magic 3rd-order squares
(2) ... (9) with the magic sums 42, 69, 96, 123, 150, 177, 204, and
231, respectively.
11
16
15
18
14
10
13
12
17
(2)
47
52
51
54
50
46
49
48
53
20
25
24
27
23
19
22
21
26
(3)
56
61
60
63
59
55
58
57
62
29
34
33
36
32
28
31
30
35
(4)
65
70
69
72
68
64
67
66
71
38
43
42
45
41
37
40
39
44
(5)
74
79
78
81
77
73
76
75
80
(6)
(7)
(8)
(9)
(2) (9) (4)
(7) (5) (3)
(6) (1) (8)
Combine these in the same order as the natural numbers in
magic square (1) to form a 9th-order magic square with the
magic sum
9 (81 + 1)
-^-^ = 369.
11
16
15
18
14
10
13
12
17
56
61
60
63
59
55
58
57
62
47
52
51
54
50
46
49
48
53
74
79
78
81
77
73
76
75
80
38
43
42
45
41
37
40
39
44
2
7
6
9
5
1
4
3
8
29
34
33
36
32
28
31
30
35
20
25
24
27
23
19
22
21
26
65
70
69
72
68
64
67
66
71

Section 5.6 Magic Squares and Kin
211
The drawing below depicts another 9th-order composite magic
square which has some interesting features besides meeting
the standard requirement that all horizontal rows, vertical
columns, and main diagonals add up to the magic sum 369:
2
6
7
8
73
72
68
67
66
11
12
13
77
78
79
81
18
59
58
57
22
19
54
71
27
30
32
49
48
46
55
70
26
35
38
43
42
47
56
69
61
51
45
41
37
31
21
5
62
53
40
39
44
29
20
4
65
36
50
33
34
52
17
3
28
23
24
25
60
63
64
1
16
76
75
74
9
10
14
15
80
By "peeling off the outer frame of numbers, we obtain a true
7th-order magic square with the horizontal, vertical, and
diagonal magic sum 287:
18
59
58
57
22
19
54
27
30
32
49
48
46
55
26
35
38
43
42
47
56
61
51
45
41
37
31
21
62
53
40
39
44
29
20
65
36
50
33
34
52
17
28
23
24
25
60
63
64
Continuing the process of removing the border of numbers, we
get, in turn, a true 5th-order magic square adding up to 205 in
all directions, and a 3rd-order square with the magic sum 123:
30
32
49
48
46
35
38
43
42
47
51
45
41
37
31
53
40
39
44
29
36
50
33
34
52
38
43
42
45
41
37
40
39
44

212
Chapter 5 COMBINATORICS
FAULHABER Johann
(1580-1635)
15-by-15 Magic Square
The magic square shown below, with 15-by-15 number cells, by
an unknown author, is from the Arithmetischer Cubic-
cossischer Lustgarten ("Arithmetic-Algebraic Pleasure
Garden"), by the German Rechenmeister Johann Faulhaber
(Tubingen, 1604). It provides an instructive example of the
general method of constructing odd -order magic squares;
there is no corresponding general rule, however, for the
construction of even-order magic squares.
We begin by entering the
number 1 in the cell
immediately to the right of
the center cell (8th row, 9th
column), and then
proceed diagonally upward
toward the right - in a
"north-easterly" diretion
- entering successively
the consecutive numbers,
in this case 2 through 7, in
the cells of the diagonal.
A diagonal that reaches
the right-hand border
column of the square is
continued at the left-hand
side of the square as if
nothing untoward had
happened; analogously,
when a diagonal reaches
the top row of the square , it
is similarly continued
in the bottom row. These
rules are illustrated by the
transitions 7-8 and 8-9,
respectively.
When a diagonal
encounters a cell that is
already occupied, and
thus obstructs the direct
continuation of the
diagonal, a re-start is made in the cell that is located two steps to the right in the
same horizontal row. This is shown by the shift 15-16, where the number 1
stands in the way of the direct progress of the diagonal.
A special case is illustrated by the cell 120 at the upper right-hand corner of the
square, where the progress of the diagonal is actually blocked by the cell 106 at
the lower left-hand corner; 120 being the last cell in the top row, the two-step
displacement of the diagonal will have to be made at the left end of the top row,
and 121 is entered in the second cell from the left. The diagonal is then
continued from cell 122 in the bottom row.
The procedures described above are repeated as often as
required until the last number - 225 - falls into the cell
immediately to the left of the center cell 113.
The sum total of all numbers in a 15th-order magic square is
n2 {n2 + 1) _ 225-226 _
2 - 2 -^4^
and the sum of every row, column, and main diagonal is
26 425
8
135
22
149
36
163
50
177
64
191
78
205
92
219
106
121
23
150
37
164
51
178
65
192
79
206
93
220
107
9
24
136
38
165
52
179
66
193
80
207
94
221
108
10
122
137
39
151
53
180
67
194
81
208
95
222
109
11
123
25
40
152
54
166
68
195
82
209
96
223
110
12
124
26
138
153
55
167
69
181
83
210
97
224
111
13
125
27
139
41
56
168
70
182
84
196
98
225
112
14
126
28
140
42
154
169
71
183
85
197
99
211
113
15
127
29
141
43
155
57
72
184
86
198
100
212
114
1
128
30
142
44
156
58
170
185
87
199
101
213
115
2
129
16
143
45
157
59
171
73
88
200
102
214
116
3
130
17
144
31
158
60
172
74
186
201
103
215
117
4
131
18
145
32
159
46
173
75
187
89
104
216
118
5
132
19
146
33
160
47
174
61
188
90
202
217
119
6
133
20
147
34
161
48
175
62
189
76
203
105
120
7
134
21
148
35
162
49
176
63
190
77
204
91
218
15
= 1695 .

Section 5.6 Magic Squares and Their Kin
213
The number magic may be extended also to other geometric
figures, for instance, to pentagrams, hexagrams, circles,
spheres, and cubes.
Magic Pentagrams and Hexagrams
In the magic pentagram shown here, the numbers are not
consecutive; the sum of the figures in every line is 28.
Of the 479 001 600 ways of arranging the consecutive numbers
from 1 to 12 at the nodes of a magic hexagram, as shown above,
there are 192 possible magic hexagrams (John R. Hendricks, "The
Magic Hexagram", Journal of Recreational Mathematics, 25:1, 1993);
the sum of the figures in every line is 26.
Magic Circles and Spheres
The magic circle shown here was designed in 1275 by the
YANG HUI Chinese Yang Hui. The sum of any circle is 138, that of any
diameter also 138 if we disregard the central 9; with it, we find
the sum 147 for all diameters.
The magic sphere, as shown above, has five great circles
(equator and four double meridians) and two small (latitude)
circles, presenting 26 crossing points in all. At these, the
numbers 1 - 26, inclusive, are entered, so that the eight numbers on
every circle add up to 108, diametrically opposite numbers
having the sum of 27.

Chapter 5 COMBINATORICS
Magic Cubes
The cube, as shown here to the left, was published at the end of
the 19th century by the Chinese Pao Chhi-shou in his Pi Nai
Shan Fang Chi ("Pi Nai Mountain Hut Records"). It has 32
numbers, while established magic cubes of the rc-th order
consist of rc3 numbers. All edges of our cube have the same sum
(41) between the vertices, which makes it, at best, semi-magic.
In a true magic cube, the sum shall be the same along all
edges, vertices included.
Assuming that the placing of the first eight numbers at the
vertices is correct, we shall change the order of some of the
other numbers to make Pao Chhi-shou's cube into a true magic
cube. We have
32
X = 528
i = l
8
I = 36
i = l
492 = 12x41
This net sum shall now be split up into twelve part sums
determined by the respective sums of the vertex pairs:
6 + 3 = 9
7 + 2 = 9
1 + 7 = 8
3 + 5 = 8
1 + 6 = 7
2 + 5 = 7
(41)
(41)
(42)
(42)
(43)
(43)
6 + 4 = 10
8 + 2 = 10
1 + 8 = 9
4 + 5 = 9
4 + 7 = 11
3 + 8 = 11
(40)
(40)
(41)
(41)
(39)
(39)
#7
Hao le
Good!
Thus, the task is to compose 12 two-component sums,
39, 39 - 40, 40 - 41, 41, 41, 41 - 42, 42 - 43, 43,
from the sequence of numbers
9, 10, 11, 12 ... 31, 32.
«SJ7
Zhao dao la I found it!
The sum along every edge
- vertices included - is 50.
©-©
7
Won le
Finished!
<§mSH5
-4jKsh

215
Chapter
6
SYMBOLIC LOGIC
Knowledge of symbols and concepts introduced in this chapter
is not needed to understand subsequent chapters. Some
concepts that originated in the field of symbolic logic are
described in the next chapter, "Set Theory".
Page
6.0 Historical Notes 216
6.1 Pitfalls 219
6.2 Propositions 220
6.3 Tautologies 225
6.4 Syllogisms and Proofs 227
6.5 Logic Circuits 229

216
Chapter 6 SYMBOLIC LOGIC
6.0 Historical Notes
ARISTOTELES
(384 - 322 B.C.)
Greek, logos, "speech",
"reasoning'
von LEIBNIZ Gottfried Wilhelm
(1646-1716)
BOOLE George
(1815-1864)
FREGE Gottlob
(1848-1925)
Predicate Logic
Aristotle, Greek philosopher and scientist - pupil of Plato, who
was a pupil and friend of Socrates - held that any logical
argument could be reduced to two premises and a conclusion,
and laid down three basic laws, or principles, of logical
reasoning, often referred to as classical logic or Aristotelian
logic:
1. The principle of identity.
A thing is itself: A is A.
2. The principle of the excluded middle.
A proposition is either true or false: either A or not A.
3. The principle of contradiction.
No proposition can be both true and false. A cannot be
both A and not A.
Aristotelian logic dominated scientific reasoning in the
Western world for 2000 years; principles of most modern
schools of logic are extensions of Aristotelian logic.
The partnership between mathematics and logic was initiated
by the German mathematician and philosopher G. W. von
Leibniz, who tried to find a lingua universalis - a language
where errors in thinking would be equivalent to arithmetical
errors - and Leibniz thereby laid the foundation for
mathematical, or symbolic, logic. A decisive stimulus to its further
development was the need at the end of the 19th century to bring
to order an abundance of collections of axioms that were being
used in various systems of mathematics in an intuitive
manner rather than by logical deduction.
The English mathematician and logician George Boole
demonstrated in his 1847 Mathematical Analysis of Logic that a
system of algebra can be used to express logical relations, and
expanded his system in An Investigation into the Laws of
Thought, on Which are Founded the Mathematical Theories of
Logic and Probabilities, published in 1854. Originally devised
as a system for logical reasoning, Boole's concepts, today
referred to as Boolean algebra, have wide applications in
electronic network design.
The German mathematician and philosopher Gottlob Frege,
founder of modern symbolic logic, constructed an elaborate
logico-mathematical system, now known as predicate logic,
which he described in 1879 in Begriffsschrift: Eine der
aritmetischen nachgebildete Formelschprache des reinen
Denkens ("Concept-Script: A Formal Language of Pure
Thought on the Pattern of Arithmetic"). Frege also wrote
Grundsetze der Arithmetik ("Basic Laws of Arithmetic"),
published, in two volumes, in 1893 and 1903.

217
DISSERTATIO
Do
ARTE COMBI-
NATOR1A,
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Do&rina'novi* jfreceptb cxftrnitur.iculu* ambarum per uni-
verfumfdcatiatumorbcm oftcndkur; nova ctiajn
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Logics Inventionis fcmina
fparguntur,
lujuu$ttSjtHffu ratfai Mbf*, & tJdtttmmt l$(»
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fcXISTENTIiE DEI,
sdMathcmattcam certitudU
ncfocxa&L*
AUTORB
GOTTrREDO GUILIELMO
LEIBNuZIO Lipscnfi,
Phil. Klagtft. U J. U. BaccaU
Afod]oh. Simom. Fickiuk at ]onu
Polyca*?, SaamoLbUM
Uteris Stom>LiANif.
A. VL PC. LX7U
Leibniz's interests in his childhood and youth did not include
higher mathematics, but rather philosophy, logic, and law,
subjects that also became the basis for his career as a diplomat.
In 1666, Leibniz earned a doctorate in law; the same year, his
interest in logic had led him to publish his first mathematical
undertaking, Dissertatio de arte combinatoria, the aim of
which was to reduce all truths of reason to a simple system of
arithmetic. Understanding that permutations and
combinations were part of such a system led Leibniz to advance also
these fields of mathematics, and that work directed him
toward ideas for the development of infinitesimal calculus.

218
Chapter 6 SYMBOLIC LOGIC
RUSSELL Bertrand
(1872 -1970)
WHITEHEAD Alfred North
(1861 -1947)
cf. p. 160
TURING Alan Mathison
(1912 -1954)
von NEUMANN John (Johann)
(1903 -1957) p. 301
The English philosopher, mathematician, and social reformer
Bertrand Russell published Principles of Mathematics in 1903
and, in collaboration with the English mathematician
and philosopher Alfred North Whitehead, the three-volume
Principia Mathematica in 1910, 1912, and 1913; carrying
on the work of Frege, they tried to derive mathematics
from self-evident logical principles. Russell and Whitehead
did not completely reach their goal, but their work has been
important for the development of logic and mathematics.
The vision - first projected by Leibniz - to reduce all truths of
reason to incontestable arithmetic was shattered in the 1930s
with Godel's incompleteness theorem, stating that any
consistent formal system adequate to describe arithmetic must
contain statements which can neither be proved nor disproved
within this system.
Today, symbolic, or mathematical, logic is used not only in
genuinely logical or mathematical domains but also in the
natural sciences, and in disciplines such as linguistics, law,
and computer technology.
A prominent figure of contemporary mathematical logic and
computer technology was the English mathematician Alan
Turing, who, with John von Neumann, opened the door to the
computer era.
Venn diagrams and Boolean algebra originated as tools of
logic; they will be described in the following chapter, "Set
Theory".
Sitaopfotaa
Georg Reisch, Margarita phy-
losophica (first edition, 1496;
here reproduced from a 1583
reprint).
The huntswoman stands on the
steady rock of Aristotelian logic.
Behind her is the Greek
philosopher Parmenides (5th
century B.C.), who, as maintained
by legend, sitting on a cliff
of the Caucasus, invented
logic. Before her is a copse of
insolubilia, and the two dogs
Veritas (Truth) and Fdlcitas
(Falsity) chasing the hare
Problema. In the background
is the forest of doctrines (silva
opinionem).
From Anders Piltz, Die Gelehrte
Welt des Mittelalters, 1982.

219
6.1 Pitfalls
Liar Paradox
Epiminides' Paradox
... further, since I had been with my master I had become aware, and was
to become even more aware in the days that followed, that logic could be
especially useful when you entered it but then left it.
Umberto Eco, The Name of the Rose (1983)
A hippie wallas down the street; friendly person remarks:
- young man, you've lost a shoe.
- Oh no, I've found one.
* * *
Ifiis statement is not true.
This is called the liar paradox or Epiminides' paradox after
Epiminides of Crete, who lived in the 6th century B.C. and,
allegedly, was the first to record paradoxes of this kind.
If the quotation is true, then it is false; if it is false, then it is
true.
* * *
The Epistle of Paul to Titus, 1.12:
One of themselves, even a prophet of their own, said, Ifie Cretians
are alway liars, evil 6easts, slow bellies.
If a Cretian says, "The Cretians are alway liars", then the
statement must include the Cretian who was quoted; though
grammatically correct, this makes no sense within the bounds
of the epistle.
* * *
The statement "I am lying" is true only if it is false, and false if it
is true.
* * *
The following story is one of many versions on the same
theme:
The prisoner was told that by making a statement, he could choose
the method of his execution; if the statement was true, he would
be shot, and if false, he would be hanged. Ifie prisoner made the
statement, "I shall be hanged".
* * *
Have you lived here all your life?
9{pj not yet.
* * *
p. 235
Russell's paradox will be described in the following chapter.

220
Chapter 6 SYMBOLIC LOGIC
6.2 Propositions
Proposition, Statement
Two-Value Principle
A proposition, or statement, describes and communicates one
or more facts; it must have a truth value.
The two-value principle encompasses only two kinds of truth
values: truth and falsity; it is assumed that no proposition is
both true and false (principle of the excluded middle).
Propositions are usually denoted by lowercase letters, such as
p, q, r, and s. In
p: 2 + 5 = 7; q: 2 + 4 = 5; r: x + 5 = 8
o proposition p is true;
o q is false;
o r is not a proposition as the equation x + 5 = 8 alone is
neither true nor false; it does not become a proposition
until we substitute a number for x.
Negation
Truth Values
Boolean Constants
Truth Table
The symbol denoting a negation is -i or, alternatively, ~ .
-ip reads, "Notp", "Nonp", or "Negation ofp".
If a proposition is false, then its negation is true.
If a proposition is true, then its negation is false.
The negation of p (x): x < 0, where x denotes real numbers, is
-i p (x): x > 0 .
If x = 5, p (x): x < 0 is false; -i p (x): x> 0 is true;
if x = - 2, p (x): x < 0 is true; -i p (x): x > 0 is false.
Statements may be expressed by the truth values, or Boolean
constants, 1 denoting truth, 0 falsity. Truth values are often
arranged as a truth table,
P "-P
1 0
0 1
Universal Quantifier
The universal quantifier V indicates "for every" or "for all".
The expression V xA,p (x) reads:
"For every x belonging to A, the proposition p (x) is valid", or
"For all x belonging to A, the proposition p (x) is valid."

Section 6.2 Propositions
221
g The expression gains in clarity if we employ the symbol g of
set theory, meaning "belonging to" or "is an element of",
x g A, p (x).
With R denoting the collection of all real numbers, the
proposition
Vjcg R, jc 2 + 1 > 0 is true;
Vjcg R, jc + 1 > 0 is false.
Existential Quantifier
3 The existential quantifier 3 indicates "there exists".
The expression 3 x A,p Gc), or 3 x e A,p (x), reads:
"There exists an x, belonging to A, for which the proposition
p (x) is valid", or
"There exists at least one x, belonging to A, for which the
proposition p (x) is valid".
With R denoting the collection of all real numbers,
3jcg R, 2 jc - 5 > 7 is true;
Zl X G K, X 2 < 0, is false.
Propositions can be combined in several ways to form a
compound proposition, or composite proposition.
Conjunctions
A The conjunction symbol a indicates "and".
The expression p a q reads, "p and q".
A conjunction is true only when both components are true:
p q p*q
ill
10 0
0 10
ooo
Given the propositions
p: 4 is an even number
q: 7 is an even number
r: 3 is an odd number
s: 5 is an odd number,
q is obviously false; p, r, and s, true. Consequently, the
conjunctions
(p at); (pa s); (r a s) are true;
(p a q); (r a q); (s a q), are false.

222 Chapter 6 SYMBOLIC LOGIC
Disjunctions
V The disjunction symbol v indicates "or".
The expression p v q reads, "p or q".
A disjunction is true when at least one of its components is
true:
p
1
1
0
0
Q
1
0
1
0
pvq
1
1
1
0
Given the propositions
p: 4 is an even number
q: 7 is an even number
r: 3 is an odd number
s: 5 is an odd number
t: 4 is an odd number
q and t are false; p, r, and s are true. As a result, the
disjunctions
q vt
t vq
qvq
tvt
qv(qvt)
tv(qvt)
have no true components and, consequently, are false; all the
other disjunctions have at least one true component, giving
them a true value.
Implications
=> The implication symbol => indicates "implies".
The expression p => q reads, "p implies q" or "if p then q".
Inp => q, we callp the antecedent, or hypothesis, and q the
conclusion:
antecedent => conclusion
p => q is false only when the antecedent is true and the
conclusion is false:
P Q P=><7
111
10 0
Oil
0 0 1
Antecedent, Hypothesis
Conclusion

Section 6.2 Propositions
223
Converse
Inverse
Logical Equivalence
Contrapositive
Principle of Detachment
The principle, or axiom, of detachment states that
ifp is true andp => q is true, then q is true:
P p=>q
1 1
The converse of p => q is q =>p.
q
1
P
1
1
0
0
q
l
o
l
o
p=>q
l
o
l
l
q=*p
l
l
o
l
The inverse of p => q is -i p
P
1
1
0
0
"■P
0
0
1
1
<7
1
0
1
0
0
1
0
1
p=>q
1
0
1
1
1
1
0
1
Two propositions are logically equivalent or, simply,
equivalent when they have true or false in the same rows of their
truth tables. The converse and the inverse of a given
implication have identical truth values and, consequently, are
logically equivalent.
The contrapositive ofp =$qis-iq =$-*p.
p => q and its contrapositive have identical truth values:
p
1
1
0
0
-■p
0
0
1
1
q
l
0
l
0
-*q
0
l
0
l
p
=>q
l
0
l
l
-*q=s
1
0
1
1
Thus, an implication and its contrapositive are logically
equivalent.
The implication x > 10 => x > 5, where x represents a real
number, is true for all real numbers x. Its converse,
x > 5 =>jc > 10,
is false; for instance, x = 8 makes the antecedent, 8 > 5, true but
the conclusion, 8 > 10, false. The inverse,
jc<10=>jc<5,
is also false, which could be predicted as the converse and the
inverse are logically equivalent.
The contrapositive ofjc>10=>:c>5,
x <5 =>jc < 10 ,
is true, which could also be predicted, as an implication and its
contrapositive are logically equivalent.

224
Chapter 6 SYMBOLIC LOGIC
<=>
Law of Substitution
The law of substitution states that in a deductive process, one
proposition may at any time be substituted for an equivalent
proposition. For instance, at any point in an argument, we
may replace the contrapositive -i g => -i p with p => g, or the
converse q =>p with its inverse -«p => -• g.
Equivalence
The equivalence symbol <=> indicates "is equivalent to".
The expression p <=> g reads, "p is equivalent to q".
p <=> g is true when the propositions have identical truth values
(both propositions are true, or both false).
p <=> g is false when the propositions do not have identical truth
values.
An equivalence has the same truth value whether read from
left to right or right to left; the equivalence p <=> q is true if and
only if p =>q and q =>p:
p
1
1
0
0
q
l
0
l
0
p
<=>g
1
0
0
1
g<=>p
1
0
0
1
The equivalence
p: x is 2 or a multiple of 2 <=> g: x is an even number,
where x denotes a real-number integer, is true for any natural
number, because whatever number we choose to represent x,
both components of the equivalence are either true or false
(identical truth values).
— Should I ivalf^ to ivorf^ or carry my (uncfi?
walk to work <=> money to buy lunch
carry lunch <=> money for transportation

225
6.3 Tautologies
"Let's talk film or let's not talk film-I'm easy"
Drawing by Victoria Roberts; © 1989
The New Yorker Magazine, Inc.
Greek, tauto, "the same thing"; A tautology is a compound proposition which is true regardless
logos, "a discourse", "a reasoning" of the truth values of its components.
The disjunction p v(-ip), an application of the principle of the
p. 216 excluded middle, is a tautology and asserts that a proposition
is either true or false - the two-value principle; there are no
values between true (1) and false (0).
p -ip pv(-ip)
10 1
0 1 1
Given the proposition
(p): The book is in the library,
the disjunction
pv(-ip),
meaning, "The book either is or is not in the library",
is a true proposition.
Given (q): a + 2 = 7, the disjunction
q v (-i q),
meaning, "a + 2 either is or is not equal to T,
is a true proposition.

226
Chapter 6 SYMBOLIC LOGIC
A proposition p a (ip) is always false, whereas its negation,
the tautology -i[p a (ip)], is always true:
p -yp pA(-ip)
10 0
0 1 0
P -ip /}A(-ip) i[pA(ip)]
10 0 1
0 10 1
The principle of contradiction, stating that a proposition and
its negation cannot be both true and false, is expressed in
symbolic form as the tautology
-i[p A(-ip)] .
Given the proposition
(p): The book is in the library,
the conjunction
P A(ip)
- meaning, "The book both is and is not in the library" - is a
false proposition.
On the other hand, the proposition
-i|p A (-ip)]
- "It is false that the book both is and is not in the library" - is
true.
Given (q)\ a + 2 = 7, the proposition
q a (-i q)
is false, but
--[g a (-. q)]
- "It is false that a + 2 both is and is not equal to 7" - is a true
proposition.
* * *
Wishful thinlqng has no use for logic - symbolic or not:
if you could choose, between your spouse and a film star, which
film star would you choose?

227
6.4 Syllogisms and Proofs
DODGSON Charles Lutwidge
English mathematician,
logician, photographer;
as Lewis Carroll, author of
children's books; 1832-1898)
"Contrariwise," continued Tweedledee, "if it was so, it might be; and if it
were so, it wouid be; but as it isn't, it ain't. cIhat's logic."
Lewis Carroll, Through the Looking-Glass (1872)
Greek, syllogismos,
"a. reasoning"
A syllogism is a valid deductive argument consisting of two
premises, usually called the major premise and the minor
premise, and one conclusion.
Msgor Premise: Only even numbers are divisible by 2
Minor Premise: 624 is divisible by 2
Conclusion: 624 is an even number
Chain Rule of Inference
Principle of Syllogisms
If the two implications p => q and q => r are true, then the
implication p => r must also be true. This may be stated
[{p => q) a (q => r)] => (p => r),
which reads, "If p implies q, and q implies r, thenp implies r".
Such an inference of conclusions from two propositions
complies with the principle, or law, of syllogisms, also called the
chain rule of inference.
p
1
1
1
1
0
0
0
0
q
l
l
0
0
l
l
0
0
r
1
0
1
0
1
0
1
0
P
=>q
1
1
0
0
1
1
1
1
q=*r
1
0
1
1
1
0
1
1
p=>r
1
0
1
0
1
1
1
1
For a direct proof, arrange a chain of implications from the
hypothesis to the theorem.
Given the true proposition p and the true implications p => q,
q => r, and r => t, prove the validity of the proposition that t is
true.
By the principle of syllogisms, we have
[(p=>q) a (q=>r)] =» (p
r).
As p is true and p => r is a true implication, we have, by the
principle of detachment, that r is true; and as r => t is a true
implication, we have, again by the principle of detachment,
that t is true. •

228
Chapter 6 SYMBOLIC LOGIC
cf. p. 159
For an indirect proof, or proof by reductio ad absurdum,
assume that the negation of the sought conclusion is true, proceed
with a chain of implications, and arrive at a contradiction.
Given the true proposition q and the true implications p =$ -*q
and -ip => r, prove the validity of the proposition that r is true.
Assume that -■ r is true.
True by hypotheses: q; p =$-*q; -*p => r
True by assumption: -■ r
As -i p => r is true, the contrapositive, -r^-i-ip, must also be
true. Using the axiom
-i -i p is true if and only if p is true,
we may state that -■ r =» p.
As -i r => p and p => -i <j are true, we have, by the principle of
syllogisms, that the proposition -■ r => -1 # is true, and thus
(-■r =»p)=»(p=»-ig)=»(ir =»ig)
is true.
The negation -ir is true by assumption and the implication
-ir =$-iq is true by proof.
If -i r were true, then, by the principle of detachment, the
negation -*q would also be true. However, q is true by
hypothesis, and by the two-value principle -*q cannot also be true;
therefore, it is a. false assumption that -i r is true.
Hence, the proposition r is true. •
Technically, there is no distinction between a direct proof and
an indirect proof. The indirect proof uses the rule
from p => q and -iq, infer -ip .
* * *
forest of Sideways Logic
After the witch was fitted, Hansel and Qretel wandered around in
the forest trying to find their way home. At a forf^ in the path
was stationed a forest ogre's guard; one path lead out of the forest,
the other to the ogre's supper table, and not as a guest!
(By order of the ogre, the guard was to be as^ed only one question by
each traveler or group of travelers. cIhe dead witch's grateful cat
had warned Hansel and Qretel that the guard lied every other time.
HanselandQretetwalked, one by one, Hansel first, then Qretel
- What question did Gretel ask the guard to be sure to infer
the correct direction?
tjpsuvycjjn nohpip uot^Tdxvp ivyfti -
The guard's statement to Gretel is true only if the statement to Hansel
was false, and false only if the statement to Hansel was true.
Consequently, if the reply to Gretel is "right", then
" right" => the road to the left is the correct one
and, if "left", then
" left * => the road to the right is the correct one.

229
6.5 Logic Circuits
Series Switch-Circuit System
Parallel Switch-Circuit System
Complex Switch-Circuit System
Graphic designs of circuits can be used not only for electric
systems but also for any system whose components constitute a
flow pattern, such as a transportation system, a computer
system, a political system, or an economic system.
A system that quits functioning if only one component is out of
order is comparable to a series switch-circuit system. The
circuit below works, because all three switches are closed:
P.
1
A parallel switch-circuit system, as shown above to the right,
functions as long as at least one component works.
A system composed of both series
and parallel networks is called
a complex switch-circuit system.
Two switches p and q in a series circuit may be expressed as
p /\q\ in a parallel circuit, p v q :
r
A switch that is always open when
another switch, p, is closed, and
vice versa, may be described as -*p
H
A switch may appear in more than one location. The same
switch can only be in one of two states: open or closed. The
switch p is both in series circuit with q and in parallel circuit
with -ip; at the same time, p must - in both locations - be
either open or closed. A closed switch has the truth value 1; an
open switch, 0:
p
1
0
1
0
-■p
0
1
0
1
Q
1
0
0
1
pAq
1
0
0
0
-■P
vq
1
1
0
1
(P
Aq) A(-ip vq)
1
0
0
0

230
Chapter 6 SYMBOLIC LOGIC
Equivalent Switch Circuits
If two switch networks have the same truth value, then they are
referred to as equivalent switch circuits:
P
Q
•<-
The above two systems have the same 1
quently, are equivalent:
P
1
1
0
0
Q
1
0
1
0
pvq (p
1
1
1
0
vq)Ap
1
1
0
0
[(P
truth values and, conse
vq)
Ap] vq
1
1
1
0
a: (p a q) v (r a t) b: (p v q) a (r v t)
c: (p v q) a\p v (q a r)] a r
Draw the corresponding switch circuits.
a:
th
c:
S.
\
By the use of symbols, describe the circuit.
{[(p a q) v (p a r)] v [q v (i r)]} a r .

231
Chapter
7
SET THEORY
Knowledge of symbols and concepts introduced in this chapter
is not needed to understand subsequent chapters.
Page
7.0 Introduction 232
7.1 Sets and Their Contents 233
7.2 Venn Diagrams 242
7.3 Algebra of Sets 249
7.4 Boolean Algebra 252
7.5 Transfinite Numbers 257

232
Chapter 7 SET THEORY
7.0 Introduction
Elements, Members
CANTOR Georg
(1845 -1918)
Set theory deals with the properties of well-defined collections,
or sets, of entities - the elements or members of the set -
conceived as a whole. The elements may be of a
mathematical nature,
{0, 1, 2, 3, 4} ; {2 x; x represents any natural number} ,
or non-mathematical,
{Robin, Kelly, Kim} ; {all humans with webbed toes} .
The mathematical theory of sets grew out of the German
mathematician Georg Cantor's study of infinite sets of real
numbers. The language of sets has become an important tool
for all branches of mathematics - as a basis for precise
definition of higher concepts, and for mathematical reasoning - but
is of very little relevance to the practice of mathematics in
everyday life.
In the 1950s educators/reformers introduced the language of
sets as the basis for mathematical studies in schools. Many
children started studying sets before they could count the
number of elements in the sets. The language of sets and the
surrounding "New Math" created havoc in schools in the
1950s, '60s, and '70s. It was a frustrating time in education.
Strange symbols were introduced for seemingly simple
things; teachers had to be retrained and most parents had no
idea what their children were doing in mathematics.
The concepts of set theory are simple, but they require a
precision and maturity of language that is beyond the power of
many students. An idea that was meant to simplify in fact
complicated matters. A dull but useful drill was replaced by a
dull and useless drill. Sadly, in Sweden and other countries,
the New Math created a generation with sometimes very
limited residual arithmetic skill.
And in high school it got even harder. There, zve had to
learn to count to 15... by heart.

233
Sets and Their Contents
Notation Example Reads
{#1, x 2 ..-%n} "Set with elements ^1,^2 • • -x n
g "belongs to" or "is an element of
x g A "x belongs to A" or "x is an
element of the set A"
£ "does not belong to" or "is not an
element of
x £ A "x does not belong to A" or "jc is
not an element of the set A"
3 "contains"
A3 x "The set A contains x as an
element"
is "does not contain"
A & x "The set A does not contain x as
an element"
Like many other notations in symbolic logic and set theory,
g was introduced, in 1889, by the Italian mathematician
Giuseppe Peano. g is a stylized form of the Greek epsilon, e,
first letter of the Greek word fccm, which means "is".
The notations g and £ ., on one hand, and 3 and &, on the other
hand, represent the same qualities seen from opposite points
of view;
if B = {a, c, k), then
a g B, c g B, and ke B, or B 3 a, B 3 c, and B 3 k
and
b £ B or B&b.
One set may be an element of another set; e.g., C = {1, 2, {3, 4}, 5}
has an element that is a set, namely, {3, 4}, which we may write
{3, 4} g C.
In the sets
A = {1, 2, 3}; B = {1, {2, 3}} ; C = {{1,2}, 3},
1 is an element of A and B but not of C,
IgA; leB: UC.

234
Chapter 7 SET THEORY
Sets of Numbers
Finite Sets
Infinite Sets
Notation
0
N
N*
Z
Z+
z_
z*
Q
Q-
Q*
R
R+
R.
R*
C
C*
Definition
0 = (}
N = {0,1,2, 3...}
N* = {1,2,3...}
Z = {.. .-2,-1,0,1,2...}
Z+(=N*) = {1,2,3...}
Z_ ={...-3,-2,-1}
Z* = {...-2,-l,l,2...}
Reads
"The empty set"
"The set of natural numbers and zero"
"The set of natural numbers"
"The set of integers"
"The set of positive integers"
"The set of negative integers"
"The set of all integers except zero
"The set of rational numbers"
"The set of positive rational numbers"
"The set of negative rational numbers"
"The set of all rational numbers except zero"
"The set of real numbers"
"The set of positive real numbers"
"The set of negative real numbers"
"The set of all real numbers except zero"
"The set of complex numbers"
"The set of all complex numbers except zero"
The exclusion of zero from the set is generally denoted by an
asterisk, as in the above N*, Z*, Q*, R*, and C*.
Sets of positive or negative numbers are generally denoted by a
subscript of a positive or negative sign, as in Z+? Z_, Q+, Q_,
R+, and R_.
The sets N, Z, Q, R, and C may be denoted by an outlined
typeface, for instance, Z, (Q, IR.
The empty set, 0, contains no elements.
A finite set is empty or contains a finite number of elements.
An infinite set contains an infinite number of elements. The
number sets N, Z, Q, R, and C are all infinite.
Examples:
Set
A = {1, 2, 3}
B = {0,2,3}
C = {0,1, {2,3}}
D = {0}
E = 0
F = {0}
G = Z
H = {Z}
Number of elements
3
3
3
1
0
1
oo
1

Section 7.1 Sets and Their Contents
235
Set Builder
Subset
Universal Set, Universe
Notation
{|} (set builder)
Also used:
{:},{;}, or[.}
Example
{x g A \p(x)}
Reads
"The set of those
elements x of A for which the
proposition p (x) is true"
If each element of a set A is contained within a set B, then A is
a subset of B. Every set is a subset of the universal set, or
universe, which contains all elements capable of being
accepted to the problem. The universal set of the problem is
usually denoted U.
If the universal set is Z, then {x e Z | x < 3} is the set of all
integers less than or equal to 3, that is, {..., — 2, — 1,0,1,2,3}.
Russell's Paradox
FREGE Gottlob
(1848 -1925)
RUSSELL Bertrand
(1872 -1970)
In 1902, in a letter to Gottlob Frege, Bertrand Russell drew
attention to the danger of unrestricted use of abstraction when
forming sets.
Suppose that there are two kinds of sets,
o Normal sets - sets that do not contain themselves as
elements; and
o Non-normal sets - sets that contain themselves as
elements.
"The set of all dogs" is a normal set, for obviously the set itself
is not a dog.
"The set of all sets" and "the set of all things that are not dogs"
are non-normal sets, for the sets are elements of themselves.
Now consider the set S whose elements x are sets that are not
elements of themselves,
S - {x | x £ x] .
Is S an element of S?
If S £ S, then S meets the requirement (x £ x) to be a
member of S and, paradoxically, S e S.
If S g S, then S fails to meet the requirement (x £ x) to be a
member of S and, paradoxically, S e S.
The contradiction,
ifS e SthenS e S; ifSe SthenS ¢5,
is known as Russell's paradox.
We may see the above as a proof of the theorem that
S - {x | x £ x) does not exist.

236
Chapter 7 SET THEORY
Shave me, shave, me not,
shave me, shave me not
shave me, shave me not,
shave me ... ta^e the sign
down.
To enact his discovery, Russell proposed, in 1918, the barber
paradox.
A barber has a sign:
I shave aCC those men in
town, andonCy those men,
who do not shave themselves.
A non-member of the set of all men in town who do not shave
themselves - a woman - can, of course, advertise as above
with impunity.
If a man, and he shaved himself, the barber would belong to the
set of men who shave themselves, but the sign implies that he
shaves only those who do not shave themselves, so he cannot
shave himself - and neither can anybody else, as he, the
barber, shaves all those men who do not shave themselves. If
he decided to wear a full beard, the contradiction prevails, for
the sign says that he shaves all those men who do not shave
themselves.
Russell's paradox was not the first paradox to be discovered in
set theory, but its simplicity and directness had an immense
impact on the development of the ideas and foundations of
mathematics at the beginning of the 20th century; before that
time the concept "class of all classes" or "set of all sets" had
been used in an unhampered manner.
Subsets
Notation
C (subset)
^ (no subset)
C (proper subset)
3 (include as a subset)
Example
B<ZA
C§ A
B(Z A
A ID B
2f (not include as a subset) A ^2f C
Z> (include properly)
J* P, or P (power set)
A Z>B
MA)
Reads
"B is included in A" or
"B is a subset of A"
"C is not included in A" or
"C is not a subset of A"
"B is a proper subset of A" or
"B is properly included in A"
"A includes B (as a subset)"
"A does not include C (as a
subset)"
"A includes B properly"
"The power set of A"

Section 7.1 Sets and Their Contents
237
The notations C, §.,, C and 3, 2> ^ represent the same
qualities seen from opposite points of view.
If A = {1, 2, 3, 4, 5}, B = {3, 5}, C = {2, 6}, and D = {4, 5,1, 3, 2},
then "5 is a subset of A" and "D is a subset of A", written
BC.A; DC.A,
or "A includes B" and "A includes D\
A^B; AiDD,
but "C is not a subset of A", or "A does not include C",
C§A; A^C;
and, as the order of the elements in a set carries no weight,
D = A.
We note that
o the empty set is a subset of every set; every set is a subset
of itself.
The correct use of the notations e , £ ("element of", "not an
element of", respectively) and C, §. is here illustrated for the
set {12}, which has one element, 12, and two subsets, 0 and the
set itself:
12 g {12}; {12}C{12}; 12§{12}
0C{12}; 0$M12}; {12} ¢{12}
We note that the notations C, C and Z> , 3 used here,
correspond to 5, C and ^,3, often encountered in other texts.
Proper and Improper Subsets
A proper subset includes elements of the set to which it is
matched, but is not equal to that set; a proper subset may also be
an empty set.
B = {2, 5} and C = 0 are proper subsets of A = {1, 2, 4, 5} .
What sometimes is called an improper subset is supposed to
include all the elements of the set to which it is matched; thus
C = {2, 1, 5, 4} would then be an improper subset of A = {1, 2, 4, 5} .
The notations C and 3 are used in connection with subsets of
a given set, whether proper or improper:
BC.A; AZ>B; C£A\ AZ>C.
Although not the practice in all texts, the symbols C and 3 are
usually preferred in connection with proper subsets,
B<ZA; AZ>B.
As the order of the elements in a set carries no weight, the very
concept of improper subsets seems unnecessary.

238 Chapter 7 SET THEORY
Power Sets
A power set is the set of all subsets of a given set, containing the
empty set and the original set.
If A = {a, b, c], then
3»(A) = (0, W, ib], {c}, {a, 6}, {a, c}, {6, c}, {a, 6, c}} .
"Power" in power sets means "exponent"; if the number of
elements contained in a set A is n, then & (A) contains 2n
elements.
Union
Notation Example Reads
u (union or "cup") AuB "The union of A
and B" or
"A union B"
. . n "The union of a
(J (union of collection of sets) (J A,- collection of sets
i = 1 A i, ... , A n
p. 222 The union notation is analogous to the disjunction notation v
of symbolic logic.
A union is a set consisting of all elements that appear at least
once in the original sets.
If B = {2, 3, 5} and C = {1, 2, 3, 6},
BuC= {1,2,3,5,6}.
n
I J A i denotes a new set formed by all elements that appear at
* = 1
least once in any of the sets A i,..., A n .
The union of a set with itself does not form a new set; if
B = {2, 3, 5},
5u5=B = (2,3,5).
• Given: The universal set R, and the subsets
A = {jcgR|-2<jc<4};jB = {jcgR|jc>0}
Find Au5.
A is the set of all real numbers x which are greater than -2 but
smaller than or equal to 4:
<>
-5-4-3-2-10 1 2 3 4 5
B is the set of all real numbers x which are greater than 0:
■O
-5 -4 -3 -2 -1 0
AuB = [xe R|jc>-2}

Section 7.1 Sets and Their Contents
239
Intersection
Notation
p. 221
Example Reads
n (intersection, or "cap") A nB
r\ (intersection of a
■ ' collection of sets)
n
nA<
/ = i
"The intersection
of A and B" or
"A intersection B"
"The intersection
of a collection of
Sets A j[, ..., j\ Yi
The intersection notation is analogous to the conjunction
notation a of symbolic logic.
An intersection is a set consisting of the elements that are
common to the original sets.
If B = {2, 3, 5} and C = {1, 2, 3, 6},
5nC = (2,3).
n
p| Ai denotes a new set formed by all the elements that are
i = l
common to all the sets A l9 ..., A n .
The intersection of a set with itself does not form a new set;
if B = {2, 3, 5},
BnB=B = {2f3f5).
Given: The universal set R, and the subsets
A = {x g R | -2 < x < 4} and B = {x e R | x> 0} .
Find An B.
A= [xe R | -2<x<4}:
<>
-5 -4 -3 -2 -1 0
B = {jc g R | x> 0}:
-5 -4 -3 -2 -1
An5 = {jcgR|0<jc<4}
0
Difference
Notation
\ (difference)
Also used: -
Example Reads
A\B
A-B
"The difference of A
and B"or
"A minus £"
The difference A \ B of sets A and 5 is the set of all elements
that belong to A but not to B. If A = {1, 2, 4, 5}, B = {2, 5}, C = {2, 3}:
A \ 5= {1,4}; A\C={1,4,5}

240
Chapter 7 SET THEORY
Complement
Notation
Example
Reads
p. 220
C (complement)
Also used:
"The complement of
subset A of IT
9 9
a\a,a
The complement notation is analogous to the negation
notation -i of symbolic logic.
Often, the notation of the universal set is not written in the
complement notation; thus, instead of Lt/A, it is usually
correct to write the complement of a subset A as Ca.
The complement is defined as the difference between the
universal set and the subset,
Cf/A = U\A.
lfU= {1, 2, 4, 5, 8, 9} and A = {2, 5, 8},
C^A = {1,4,9}.
While the definitions of "union" and "intersection" do not
require specification of the universe, the definition of
"complement" does.
MatchSuCtf; A u Ca ; B n£,B ; A oCa ; C( C#); C( Ca)
with 0, u, a b, Ca, Cs,
where A and B are subsets of the universal set U.
Bv£b = U; AuCA=t/; Bn^B = 0; AnCA = 0;
C(O) =5; C(Ca) = A.
Given: The universal set R, and the subset
A = [x g R | -2 < jc < 4} .
Find CRA.
0
The complement of A contains -2, all real numbers smaller
than -2, and all real numbers greater than 4 (but does not
include 4),
c
R
A = [x | x < -2 or x> 4} .

Section 7.1 Sets and Their Contents
241
Ordered Components and Product Sets
In an ordered pair (a, 6), or an ordered n-tuplet (al9a2,..., a n\
the order is of significance:
(a, 6) = (6, a) only if a = 6
The elements within an ordered pair or an ordered n-tuplet are
Components called components. In (a, 6), a is the first component and b is
the second component.
If sets are combined into sets of ordered pairs the result is
Product Sets, Cartesian Products called product sets or Cartesian products. To indicate the
forming of a product set, the symbol x is used.
The product sets of A = {1, 2, 3} and B = {a, 6} are
A x B = {(1, a), (1, 6), (2, a), (2, 6), (3, a), (3, 6)}
B x A = {(a, 1), (a, 2), (a, 3), (6, 1), (6, 2), (6, 3)}
A x A = {(1,1), (1, 2), (1, 3), (2, 1), (2, 2),
(2, 3), (3, 1), (3, 2), (3, 3)}
BxB = {(a, a), (a, 6), (6, a), (6, 6)}

242
Chapter 7 SET THEORY
7.2 Venn Diagrams
von LEIBNIZ Gottfried Wilhelm
(1646-1716)
EULER Leonhard
(1707-1783)
VENN John
(1834-1923)
A Venn diagram is a rectangle - the universal set - that
includes circles depicting the subsets.
The first to systematically use diagrams to represent
statements of logic reasoning was the German mathematician and
philosopher G. W. von Leibniz. The diagrams used today are
sometimes referred to as Euler diagrams after the Swiss
mathematician Leonhard Euler, who devised them, but are usually
called Venn diagrams after the English logician John Venn,
who, in 1880, greatly improved the diagrams and popularized
their use.
A Venn diagram can be employed for any number of subsets,
but more than three defeat the purpose of gaining increased
clarity.
Two Subsets
Union. A and B are subsets
of the universal set U. The
shaded area represents
AuB.
Intersection. A and B are
subsets of the universal
set U. The shaded area
represents AnB.
Complement. A and B are
subsets of the universal
set U. The shaded area
represents \jA.
Complement. A and B are
subsets of the universal
set U. The shaded area
represents \jB.
Union of Complements. A
and B are subsets of the
universal set U. The shaded
area represents L L
Intersection of
Complements. A and B are subsets
of the universal set U. The
shaded area represents

Section 7.2 Venn Diagrams
243
Intersection of
complement and subset.
A and B are subsets of the
universal set U. The
shaded area represents
(CA)nB.
Union of complement and
subset. A and B are subsets
of the universal set U. The
shaded area represents
(Ca)uB.
Complement of union of
subsets. A and B are subsets
of the universal set U. The
shaded area represents
C(AuB.
QD
Intersection of subset
and its complement.
A and B are subsets of the
universal set U.
Bn^B = 0.
Union of subset and its
complement. A and B are
subsets of the universal
setU. Bu C#.
Disjoint sets. A and B are
subsets of the universal set U.
A and B have no elements in
common. Sets that have no
elements in common are
called disjoint sets.
Proper subset. A and B are
subsets of the universal set U.
A is a proper subset of B, which
may be expressed A C B.
Also, e.g.,
AuB = BandAnB = A.
Match each of
AuB, AnB , An C B,
Caub, CapiCb, CauC#
with one of
0, A, B, U, ZB, C A .
A\£,
AkjB=B; AnB = A; An[,B = 0; A\B = 0;
CAuB =U; CAnC#=C5; CAuC#=Ca.

244
Chapter 7 SET THEORY
Construct Venn diagrams showing
a:An£ = 0 b: Au5 = A c:A\B=A,
where A and B are subsets of the universal set U.
a:
b:
c:
00
Given: U = {1, 2, 3, 4, 5, 6, 7, 8, 9,10}
A = {1, 2, 4}
5 = {4, 5, 6, 7}
Display the sets in a Venn diagram and determine
a:Au£ b: AnB c:A\B d:B\A
e: CA f: Zb g: C(Au5) h: C(An5)
a: A u5 = {1, 2, 4, 5, 6, 7} f: C# = {1,2, 3,8, 9,10}
b: AnB = {4} g: C(Au£) = {3,8,9,10}
c: A \ B = {1, 2}; h: C(An£) = {1, 2, 3, 5, 6, 7, 8, 9,10}
d: B \ A = {5, 6, 7}
e: CA = {3, 5, 6, 7, 8, 9,10}
Of 160 individuals with a skin disorder, 100 had been exposed
to chemical A (individuals A), 50 to chemical B (individuals
B), and 30 to chemicals A and B.
Use symbols and Venn diagrams to describe the number of
individuals exposed to
a: chemicals A and B
b: chemical A but not chemical B
c: chemical B but not chemical A
d: chemical A or chemical B
e: neither chemical A nor chemical B

Section 7.2 Venn Diagrams
Solution:
Of individuals A,
100 - 30 = 70
had been exposed only to
chemical A.
Of individuals B,
50-30 = 20
had been exposed only to
chemical B.
a:
The number of individuals
exposed to both chemical A
and chemical B can be
expressed as
\AnB\ = 30.
b:
The number of individuals
exposed to chemical A but
not to chemical B can be
expressed as
|AnC#| = 70.
c:
The number of individuals
exposed to chemical B but
not to chemical A can be
expressed as
| CAn£| = 20.
d:
The number of individuals
exposed to chemical A
or chemical B can be
expressed as
\AuB\ =120.
e:
The number of individuals
exposed to neither chemical
A nor chemical B can be
expressed as either
| CAnC5| =40
or | C(Au£)| = 40.

246
Chapter 7 SET THEORY
Three Subsets
When three circular areas are used to represent three subsets
of the universal set and the areas overlap, eight regions may be
formed:
A, B, and C are subsets of the
universal set U. The shaded
area represents AnB nC.
u
jfyyyyy^-yyyyyS^
■f^yyyyyyy\-yyjryyyyyyy.
yy\
&/
A, B, and C are subsets of the
universal set U. The shaded
area represents AuBuC.
A, B, and C are subsets of
the universal set U. The
shaded area represents
C(AuBuC.
U
A, B, and C are subsets of the
universal set U. The shaded
area represents
C(AuC)n£.
800 individuals were examined for antigens called <xi, 0C2, and
oc3.
Number of individuals positive for
«i
«2
«3
oci and 0C2
0C2 and 0C3
oci and 0C3
<xi, 0C2, and 0¾
The number of individuals negative for all
three antigens, <xi, 0C2, and a3.
The number of individuals positive for antigen
0C2, but negative for both oci and 0C3.
500
350
400
250
150
200
50
Find a

Section 7.2 Venn Diagrams
247
Solution:
Construct Venn diagrams with three circles and let
circle A hold elements representing individuals positive
for antigen oci (500);
circle B, individuals positive for antigen oc2 (350);
circle C, individuals positive for antigen 0C3 (400):
1. 50 individuals are
positive for all three
antigens.
2. 200 individuals are
positive for oci and 0C3;
of these individuals, 50
are already displayed in
the figure to the left, and
150 remain.
u
\a
\i 50/vi ooy
B J
3. 150 individuals are
positive for 0C2 and 0C3;
of these individuals, 50 are
already displayed in the
above, and 100 remain.
5. The remaining elements
of region A are
500 - (50 + 150 + 200) = 100 .
The remaining elements of
region B are
350 - (50 + 100 + 200) = 0 .
The remaining elements of
region C are
400 - (50 + 100 + 150) = 100 .
4. 250 individuals are
positive for ax and 0C2; of these,
50 are already displayed in
previous figures, and 200
remain.

Chapter 7 SET THEORY
The number of individuals negative for all three antigens oci.
0C2, and 0:3 is C(Au5uC).
Since the universal set is 800, and | A u B u C | = 700, we have
I C(Au5uC)l = 800-700 = 100,
which is the answer to a.
We can now complete the diagram:
The number of individuals positive for antigen 0C2 but negative
for oci and 0:3 is B n L(A u C).
The above diagram shows that the set of individuals positive
for antigen 0C2 but negative for oci and 0C3 is empty; thus,
BnC(AuC) = 0,
which is the answer to b.
a: 100 of the 800 examined individuals have no antigen oci, 0C2,
or 0C3 .
b: No individuals positive for antigen 0C2 are negative for oci
and a3 (that is, all individuals with antigen a2 also have
either oci or 0C3, or oci and 0C3).
* * *
oo>
IB
C0 33
ffiflp
<&&
QQesca?^
HLD
Q3
A shoemaf^er died, leaving behind 17 pairs of men's shoes and boots.
In his Cast will and testament he disposed of them as follows: 1/2
of the collection to his youngest, hardworlqng, son; 1/3 to the
middle son-, and 1/9 to "Lazybones", the eldest son.
Understanding that it was not possible to divide the 17 pairs into
halves, thirds, and ninths, the exasperated executor of the will
tossed in his own footwear, mating it a total of eighteen pairs,
which he divided as follows: 1/2, or 9 pairs, to the youngest son;
1/3, or 6 pairs, to the middle son; and 1/9, or 2 pairs, to the eldest
son.
Upon realizing that adding up 9, 6, and 2 made 17, the executor
elatedly retrieved his own pair of shoes and closed the estate.

249
7.3 Algebra of Sets
Fundamental Laws
There are similarities but also significant differences
between conventional algebra and the algebra of sets.
Commutative Laws
1. Au5 = BuA
2. AnB = BnA
These identities follow directly from the definition of union
and intersection:
Au5=5uA AnB = BnA
Idempotent Laws
1. AuA = A
2 AnA = A
Associative Laws
LAu(5uC) = (AuB)uC
2. An(BnC) = (AnB)nC
The associative laws of set theory follow directly from the
definitions of union and intersection:
Au(5uC) = (AuB)uC AniBnC) = (AnB)nC

250 Chapter 7 SET THEORY
Distributive Laws
1. Au(BnC) = (Au5)n(AuC)
2.An(BuC) = (A nfl) u (A n C)
These laws can be proved by successively building on to the
Venn diagrams of the expressions on either side of the equality
sign.
For instance, to prove A u (B n C) = (A u B) n (A u C), we
have:
Left Side Right Side
Au(BnC) (A u 5) n (A u C)
Identities Involving Empty Sets
1. A u 0 = A
2. A n 0 = 0
de Morgan's Laws
1. C(AuB) =(CA)n C#
2. C(An5) =(CA)u C#
These laws, named after the British logician and mathemati-
de MORGAN Augustus cian Augustus de Morgan, read, respectively:
(1806-1871)
o The complement of the union of two sets is the
intersection of the complements.
o The complement of the intersection of two sets is the
union of the complements.

Section 7.3 Algebra of Sets
251
de Morgan's laws can be proved by successively building on to
Venn diagrams of the expressions on either side of the equality
sign.
For instance, to prove B, we have:
Left Side
Au5
Right Side
Ca
Cb
C
(AuB)
|t'"Uiiijiinn
r --^-->■■>■ ■■>/■'
npwfPWPTPnpvvvwi
Ca^> c
To demonstrate the unzuieCdiness of Venn diagrams for depicting
more than three subsets, the instructor brought his pet elephant

252 Chapter 7 SET THEORY
7.4 Boolean Algebra
Boolean algebra is concerned with ideas or objects that have
only two possible stable states - e.g., on/off, closed/open,
yes/no, true/false.
This algebra was devised by the English mathematician and
BOOLE George logician George Boole in 1847, but it was not until the middle of
(1815 -1864) p. 217 the 20th century it was applied to the analysis of relay and
switching circuits.
Notations and Definitions
Notations previously used in this chapter may be used in
Boolean algebra, but it is customary to use:
' or ~" for negation or complement ( L ) a'; a
© or + for union (u) a © 6; a + b
(8>, •, or x for intersection (n) a (8>b; a - b; axb
We favor the notations ', ©, and (8>; criticisms of this use could
be raised, however, as © and <8> are established notations for
direct sum and tensor product, respectively.
a' is the complement of a.
The results of operations © and (8) are referred to as sum and
product, respectively.
As in conventional arithmetic and algebra, parentheses
indicate precedence of operations; otherwise, complement (')
has precedence over (8), and (8) has precedence over ©. To avoid
confusion, we will be generous with the use of parentheses.
A Boolean set is commonly referred to as B.
0 and 1 denote two distinct identity elements of B. 0 is the zero
element; 1 is the unit element. The zero element of a Boolean
algebra corresponds to 0 of general set theory; the unit
element corresponds to the universal set.
Fundamental Laws
Commutative Laws
cf.p.249 a®b = b®a a®b = b®a
Identity Laws
a©0 = a a(8>l = a
The set a © 0 is the set of elements belonging to either the set a
or to the empty set 0. This set is identical with the set 0. This
set is also identical with the set a, so a©0 = a.
a (8) 1 = a, because set a is contained in the universal set;
consequently, the elements common to both sets are precisely
the elements of set a.

Section 7.4 Boolean Algebra
253
Distributive Laws
cf. p. 250 a © (6 <8> c) = (a © 6) (8) (a © c) a (8) (6 © c) = (a <8> 6) © (a (8) c)
Complement Laws
A statement a is either true or false; therefore,
a®ax = 1.
A statement a cannot be both true and false; therefore,
a®ax = 0.
Supplementary Laws
Supplementary Complement Laws
1. 0' = 1 1' = 0
2. (a,y = a (involution law)
3. Ifa©:c = landa(8>:c = 0, thenjc = a'
Boundedness Laws
a©l = 1 a<8>0 = 0
Using the terms of general set theory, equation a © 1 = 1 states
that the union of elements contained in set a or the universal
set (1) cannot exceed what is already contained in the
universal set.
Equation a (8) 0 = 0 implies that there are no elements common
to set a and an empty set; that is, a set formed by elements
belonging to set a and an empty set is an empty set.
Duality Principle
By interchanging the operations © and (8) and interchanging
the identity elements 0 and 1 of a valid expression, we obtain
another valid expression, called the dual of the original
expression.
The dual of (1 © b) (8) (a © 0) = a is
(0 (8) b) © (a (8) 1) = a .
Absorption Laws
a © (a (8) b) = a a (8) (a © b) = a
Associative Laws
cf. p. 249 (a © 6) © c = a © (6 © c); (a (8) 6) (8) c = a (8) (6 (8) c)
Idempotent Laws
A product x (8) jc, sometimes written jc2, is the set which has the
properties of set x; therefore,
x (8) x (= jc2) = jc .

254 Chapter 7 SET THEORY
Similarly,
x <8> x <8> x (= jc3) = x,
x (8) x (8) jc (8) jc (= x4) = jc, e£c.;
and
•A* xL/ ^v "~" ^v %
JC © JC © X = JC,
jc © jc © jc © jc = jc, e£c.
de Morgan's Laws
c/!p.250 (a©6)' = a'<8>6' ; (a (8) 6)' = a'© 6'
Simplifying Boolean Expressions
It often requires considerable patience, ingenuity, or luck to
simplify a Boolean expression.
Examples:
o x' © (y (8) x) can be simplified to x' © y by the
distributive law,
*'© (y <8>jc) => (jc'©y)<8> (*' © jc) ;
complement law,
(xx ©y) (8)(jc'©jc) => Oc'©y) <8> 1 ; and
identity law,
0c'©y)<8> 1 => x'©y .
o x (8) (jc © y) can be simplified to x by the
identity law,
x (8) (jc © y) => (jc © 0) (8) (x © y) ;
distributive law,
(x © 0) (8) (x © y) => x © (0 <8> y) ;
boundedness law,
x©(0<8>y) => jc©0; and
identity law,
x © 0 => jc .
o (jc © y)' © (x (8) y') can be simplified to y' by
de Morgan's law (a © 6)' = a' (8) 6' ,
(jc©y)'©(:c <8>y') => (*'<8>y')©(jc <8>y') ;
W <8>y') © (jc <8>y') => (y'<8>jc') © (y'<8>x) ;
the distributive law,
{yx (8) jc') © (y' <8> x) => y' (8) (*' © jc) ;
the complement law jc' © x = 1, and thus,
y'(8)(jc'©jc) => y'(8> 1 ;
and, finally, the identity law,
y'<8>l => y'.

Section 7.4 Boolean Algebra
255
Busting the Brainbuster
Problems like the one discussed here are frequently
encountered in "brainbusters". Boolean algebra may offer a reliable
approach, but simple logic or common sense more often
provides the optimal solution.
A congeniality contest in the forest has four prizes, 1st, 2nd,
3rd, and 4th. Three judges (I, II, III) announce the
prizewinners (A, B, C, D) by each giving one true and one false:
I
II
III
1st
A
2nd 3rd
B
A D
C
4th
D
A =
B =
C =
D =
Find the order of the prizewinners.
First, a solution suitable for an automaton:
While, in a Boolean algebra, a disjunction (a © operation) is
true if at least one of its components is true, a conjunction (a ®
operation) is true only if both components are true. Therefore,
for the announcements of, for instance, judge I, we have
A1®B2 = 1; A1®B2 = 0,
where the indices indicate the announced order of the
prizewinners.
If the conjunction of announcement A\ and the complement of
B2 is true, then the conjunction of announcement B2 and the
complement of A\ is false, and vice versa; therefore,
A1®(B2y®B2®(A1y = l.
Similarly, we have for the announcements of judge II,
A2®(D3y®D3®(A2y = 1;
and forjudge III,
c2®(D4y®D4®(c2y = l.
The conjunction of any number of true propositions is a true
proposition; therefore,
[A! ® (B2y ®B2® (Ai)'] ® [A2 ® (D3y ®DS® (A2)']

256 Chapter 7 SET THEORY
The process of multiplying out the left-hand side of the
equation
[Ai <8> (B2r ®B2® (Ai)'] <8> [A2 <8> (¾)1 ®D3® (A2)']
<8>[C2<8>(D4)'0D4<8>(C2)'] = 1
is more easily visualized if instead of (8) we use juxtaposition;
thus,
[Ai (B2Y 0 B2 (Ai)'] [A2 (D3)' © #3 CA2)'] [C2 (¾)1 © #4 (¾)1 ]
= A! (B2)'A2 (/)3)-02 (/)4)1 SAj (£2)^3(^2)^4(^2)1
ea1 (B2y d3 (A2yc2 (D4y ®a1 cb2)'a2 (d3)' #4 (c2y
0 £2 (Ai)'A2 (D3)' C2 (D4y 0 £2 (Ai)'D3 (A2)'D4 (C2)'
©BaCAiVDsCAayCa^)' e52(A1),A2 (D3)'D4(C2)' = 1.
The /irs^ term is false (0), for A cannot be both 1st and 2nd; the
second is false, for D cannot be both 3rd and 4th; the fourth is
false, for A cannot be both 1st and 2nd; the fifth is false, for A
and B cannot both be 2nd; the sixth is false, for D cannot be both
3rd and 4th; the seventh is false, for B and C cannot both be
2nd; the eighth is false, for B and A cannot both be second.
We are left with the third term,
A! (£2)^3 ^2)^2(2)4)1 = *'
which reads: A is 1st, B is not 2nd, D is 3rd, A is not 2nd, C is
2nd, D is not 4th.
The order of the prizewinners is:
Hedgehog (A) - Rabbit (C) - Squirrel (D) - Fox (B)
Machines do not take shortcuts; human beings do!
=>, "implies" (p. 222) | B2=$ D3 and D4, so B2 is false and Ai is correct; then,
Ai => D3, so B4 and C2 .
- Child's flay, my dear Robert.

257
Transfinite Numbers
Since ancient times the subject of infinity has occupied the
minds of men and women of philosophy, theology, and
mathematics. Today the concept of infinity becomes especially
intriguing when we compare the infinitude of all natural
numbers, all even numbers, all odd numbers, all rational
numbers, all irrational numbers, all real numbers, etc. Then
the question arises:
Are there different sizes, or degrees, of infinity?
Galileo Galilei, the Italian mathematician, astronomer, and
physicist, argued in his Dialogue Concerning the Two Chief
World Systems (1632):
There are as many squares as there are naturat numbers
because they are just as numerous as their roots.
In view of the fact that there are natural numbers that are not
squares - e.g., 2, 3, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 17 - Galilei's
statement seems to be a paradox.
The question of infinity and the possibility of different degrees
of infinity was given new life when the German
mathematician Georg Cantor, the inventor of set theory, in 1874
presented a method of investigating the concept of infinity in
Uber eine Eigenschaft des Inbegriffes alter reellen Zahlen
("On the Characteristic Property of All Real Numbers").
Cantor published several other mathematically revolutionary
papers on set theory and infinity, culminating in 1895 - 97 in
his best-known work, Beitrdge zur Begriindung der trans-
finiten Mengenlehre, published in Matematische Annalen; a
French translation appeared in 1899; an English
translation, by Philip E. B. Jourdain, in 1915: Contributions to the
Founding of the Theory of Transfinite Numbers.
Although mainly examining infinite sets, Cantor developed
most of his ideas from concepts of finite sets. Two finite sets
contain the same number of elements when there is an exact
correspondence between the two sets. For example, there is a
one-to-one correspondence between six people and six chairs
around a dining table, between the fingers of the hands, or
between the fingers of one hand and the toes of one foot.
Cantor showed that the concept of one-to-one correspondence
can be as readily applied to infinite sets as to finite sets.

258 Chapter 7 SET THEORY
Denumerably Infinite Sets
Sets that are equivalent to the set of all natural numbers
are called denumerably infinite sets. Thus, the members of a
denumerably infinite set can be put in one-to-one
correspondence with the infinitude of natural numbers {1, 2, 3, ...}, as
shown in the following examples.
Our "finite intellects" tell us that the number of natural
numbers ought to be larger than the number of even numbers.
However, to prove the proposition that the set of all natural
numbers and the set of all positive even numbers are
denumerably infinite sets, we need only make the following
one-to-one correspondence diagram:
1 2 3 4 5 6 7 ... 7i ...
2 4 6 8 10 12 14 ... 2n ...
In the same way, the sets of all odd numbers can be shown to be
denumerably infinite:
1 2 3 4 5 6 7 ... n
1 3 5 7 9 11 13 ... 2n-l ...
By making a one-to-one correspondence diagram, we can
prove Galilei's statement that there are as many squares as
there are natural numbers:
123456789...
12 22 32 42 52 62 72 82 92 ...
In the same way we can show that the sets of all cubes, natural
numbers to the fourth power, fifth power, etc., are denumerably
infinite.
In the above examples the set of the infinitude of natural
numbers is put in a one-to-one correspondence with a subset of
itself: the whole is equal to part of itself. This is a glaring
contrast to the axiom of finite magnitudes, where the whole is
equal to the sum of its parts and, therefore, is greater than any
of them.
We may, with Cantor, define an infinite set as one that can be
put in one-to-one correspondence with a proper subset of itself.
This had been proposed in 1872 by the German
mathematician Richard Dedekind in Stetigkeit und irrationale Zahlen
("Continuity and Irrational Numbers"), but Cantor realized
that not all infinite sets are the same.
Cantor denoted the manyness of elements of a denumerably
n0 infinite set with the cardinal number N0> which reads "aleph-
null", "aleph-zero", or "aleph-naught"; N is the first letter of
the Hebrew alphabet. Larger infinite sets - those that are not
in one-to-one correspondence with the infinitude of natural
numbers - are denoted Rl9 ^2, ^3» etc.
DEDEKIND Richard
(1831-1916)

Section 7.5 Transfinite Numbers
259
Latin, trans, "across", "beyond";
finire, "to end"
N0, Nx, N2> ^3, etc., are the cardinal numbers of infinite sets,
where each successive set has a higher degree of infinity; such
cardinal numbers are called transfinite numbers.
Cantor proved that the set of all integers, the set of all natural
numbers, the set of all rational numbers, and the set of all
algebraic numbers are denumerably infinite.
As one can interpose an infinity of rational numbers between
any two given rational numbers, one would be tempted to
believe that the set of all rational numbers is non-denumer-
able. However, by arranging the set of all rational numbers
as below, we can prove that the set of all rational numbers is
denumerable and, thus, has the cardinal number N0 •
0
1/1 = 1
1/2
1/3
1/4
1/5
-1/1=
-1/2
-1/3
-1/4
-1/5
-1 2/1 = 2
2/2 = 1
2/3
2/4 = 1/2
2/5
-2/1 = -2
-2/2 = -1
-2/3
-2/4 = -1/2
-2/5
3/1 = 3
3/2
3/3 = 1
3/4
3/5
-3/1 = -3
-3/2
-3/3 = -1
-3/4
-3/5
4/1 = 4 ...
4/2
4/3
4/4
4/5
In the first column all numerators are 1, in the second -1,
third 2, fourth -2, etc.; in the first row, after 0, all
denominators are 1, in the second 2, third 3, etc.
Delete all fractions that have a factor common to the
numerator and he denominator; every rational number will
now appear only once:
4
4/2
4/3
4/4
4/5
3
3/2
3/4
3/5
-3
-3/2
-3/4
-3/5
The arrows indicate the order in which we set up a one-to-one
correspondence between the natural numbers and the rational
numbers so that no rational number will be omitted:
1
u
0
2
u
1
3
u
1/2
4
u
-1
5
u
2
6
u
-1/2
7
u
1/3
8
u
1/4
9
u
-1/3
10
u
-2
11
u
3
12
u
2/3
13
u
-1/4
14
u
1/5
Thus, with Cantor, we have proved that the set of all rational
numbers is denumerably infinite.

260 Chapter 7 SET THEORY
Non-Denumerably Infinite Sets
Cantor also showed that there are transfinite numbers that
are larger than N0. Specifically, he proved that the set of all
real numbers is non-denumerably infinite; that is, there is no
one-to-one correspondence between the set of all real numbers
and the set of all natural numbers.
The proof of the non-denumerability of the set of all real
numbers is necessarily indirect. For the proof, we need only
an interval of real numbers; we choose the interval that is
strictly between 0 and 1.
For the proof, write all numbers to be considered in the form of
infinite decimal expansion, e.g.,
1 = 0.999 9...; 0.367 = 0.366 999 9...; 0.001 = 0.000 999 9...
and assume that the infinite decimal expansion of all real
numbers between 0 and 1 can be put in one-to-one
correspondence with the set of all natural numbers, N* = {1, 2, 3 ...}:
N*
1
2
3
4
5
6
7
•
•
n
m
m
m
=»
=»
=>
=»
=>
=»
=>
=>
decimal expansion of all real numbers between 0 and 1
0.
0.
0.
0.
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Consider a number
U. ZjZjj £jjj Z jy ...
such that
0 2ris 8 if a\ is 0, 1, 2, 3, 4, 5, 6, or 7 ; zl is 1 if a\ is 8 or 9
0 zu is 8 if 62 is 0, 1, 2, 3, 4, 5, 6, or 7 ; zu is 1 if 62 is 8 or 9
0 zul is 8 if C3 is 0, 1, 2, 3, 4, 5, 6, or 7 ; zm is 1 if C3 is 8 or 9
etc.
The number 0. ZiZuzmzlv ... cannot be equal to any decimal
expansion of any row in the diagram. Thus, there is no
equality between 0. zl zu zlu zlv ... and 0. a\a<La%a± ..., for
by design zx differs from a\, or between 0. zlzllzmzlN ... and
0. 61 62 ^3 ^4 • • •> f°r z\\ differs from 62» e^c-
Thus, it is a false assumption that the infinite decimal
expansion of all real numbers between 0 and 1 can be put in one-to-
one correspondence with the set of all natural numbers; the
cardinal number of the set of all real numbers lying between 0
and 1 is greater than N0.

Section 7.5 Transfinite Numbers
261
Cantor also proved that there is an infinitude of other infinite
sets whose size is larger than the set of all natural numbers.
Although the above investigation shows that the cardinality of
the set that contains all real numbers between 0 and 1 is
greater than K0, it is not possible to tell whether this greater
number is K1? K2, N3, or any other transfinite number greater
than K0.
Algebra of Transfinite Numbers
Being an algebra of sets, the algebra of transfinite numbers
offers no real surprises at this stage:
No + No = N0
K0 + K0+... + N0 = N0
N0xN0 = N0
K0x K0 x ... x K0 = Nq
Transfinite numbers and transfinite induction have been
applied in the form of "non-standard analysis" for proofs of
results associated with the Boltzmann equation in kinetic gas
theory.
... sed libera pueros ludi a Cantu Cantoris.
"... and deliver our small schoolchildren from set theory."

262
Chapter 7 SET THEORY
The Continuum Hypothesis
GODEL Kurt
(1906 -1976); p. 160
COHEN PaulJoseph
(b. 1933)
The transfinite number expressing the cardinality of the set
that contains all real numbers between 0 and 1 is denoted c.
The letter "c" stands for continuum and refers to Cantor's
continuum hypothesis.
According to the continuum hypothesis, any infinite subset
of real numbers either is denumerable, that is, in one-to-one
correspondence with the set of natural numbers, or can be put
in a one-to-one correspondence with the set that contains all
real numbers between 0 and 1.
Cantor's efforts to prove the continuum hypothesis were
fruitless. In 1940, Kurt Godel showed that the continuum
hypothesis cannot be disproved, and in 1963 Paul J. Cohen, an
American mathematician with an interest in logic, showed
that the continuum hypothesis is, in fact, undecidable: besides
true statements and false statements, set theory also has
undecidable statements.
* * *
MITTAG-LEFFLER Gosta
(1846 -1927)
During a time when editors were generally unsympathetic to
articles about transfinite numbers, Swedish mathematician
Gosta Mittag-Leffler - founder and editor of Acta Mathemat-
ica - took great interest in Cantor's discoveries and, thus,
much of Cantor's work was published in Acta Mathematica.
Inscribed on the mantelpiece in Gosta Mittag-Leffler's home
- now a research institute for mathematics - in Djursholm,
Sweden, we read:
Tafet iir tdnkandets Sorjan ocft slut.
Medtauten foddes tafet
Utofver tafet nar tauten icfe.
ML 1903

263
Chapter
8
INTRODUCTION
TO SEQUENCES AND SERIES
8.1
8.2
8.3
8.4
8.5
8.6
*** 22
Terminology
Finite Sequences and Series
Infinite Series
The Tower of Hanoi
The Fibonacci and Related Sequences
Figurate Numbers
Power Series
Page
264
266
270
285
286
289
765
*** Indicates cross-reference

Chapter 8 INTRODUCTION TO SEQUENCES AND SERIES
8.1 Terminology
Sequence, Progression
Term, Element
General Term
Series
Alternating Series
Arithmetic Sequence
Common Difference
Arithmetic Series
Harmonic Sequence
Harmonic Series
Geometric Sequence
Common Ratio
Geometric Series
A sequence or progression of numbers is a set of numbers
arranged in an orderly fashion such that the preceding and
following numbers are completely specified; in the sequence
...,7,10,13,16,19,...
the difference between numbers is 3, which indicates that the
number preceding 7 is 4 and the number following 19 is 22.
The numbers of a sequence are called terms or elements. The
general term defines the rule of the sequence; if n is the
ordinal number of a term in the sequence, 1 is the first term, and
the general term an is 2n - 1, the sequence is
^•4 ^-'j *~^? I a • • ma £j /if ^* 9 • • • a
if the general term an is n2, the sequence is
14 9 n2
^. a 7a KJ a • • • a ft/ a * • • *
A series is the sum of the terms of a sequence,
Unlike a series of positive terms, such as
5
I
n = 0
n
^ 11111
= 1 +~ + ~ + ~+ — + —
24816 32'
an alternating series has alternately positive and negative
terms,
5
f-iY
i
n = 0
1 1 _ 1 J__ J_
"2 + 4"8+16"32 *
In the arithmetic sequence 1, 3, 5, 7, 9, 11 ... the terms have a
common difference of 2 units; the corresponding series is an
arithmetic series.
A sequence whose terms are the reciprocals of an arithmetic
sequence is a harmonic sequence; the corresponding series is
a harmonic series. The terms of the harmonic sequence 1, 1/2.
1/3, 1/4, ... are the reciprocals of the terms of the arithmetic
sequence 1, 2, 3, 4, ....
In the geometric sequence 1, 2, 4, 8, 16, 32, ... the terms have
a common ratio of 2 units; the corresponding series is a
geometric series.

Section 8.1 Terminology
265
Finite Sequences and Series
Infinite Sequences and Series
Convergent Series
Divergent Series
Finite sequences and series have defined first and last terms,
e.g.,
1 + 3 + 5 + 7 + 9+11 + 13 + 15;
infinite sequences and series continue indefinitely,
1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + ....
Infinite series whose terms increase in magnitude have no
attainable sum.
For an infinite series to have a sum its terms must get ever
smaller, but this is not the only requirement; a series whose
terms decrease in magnitude toward zero may have a sum or
may have no attainable sum.
If an infinite series has a finite sum, it is referred to as a
convergent series; we say that the series converges. If it has
no sum, it is known as a divergent series; it diverges.
We note that, in mathematical terminology, the expressions
"convergent" and "divergent", when applied to a series, refer
exclusively to the existence of a sum or no sum.
^
4^¾^

266 Chapter 8 INTRODUCTION TO SEQUENCES AND SERIES
8.2 Finite Sequences and Series
Arithmetic Sequences and Series
A finite arithmetic sequence may be written
ai, a\ + d, a\ + 2d, a\ + 3d, ..., a\ + (n-l)d,
where a\ denotes the first term, n is the number of terms, and d
is the common difference.
In an arithmetic sequence or series, the rc-th term is
an = a i + in - l)d .
• Find the 17th term and the last term of the arithmetic series
52 + 56 + 60 + ... + a 32.
The common difference is
56-52 = 4,
so
a17= 52 + (17-1)4 = 116
and
a32= 52 + (32-1)4 = 176.
Harmonic Sequences and Series
To find a specific term of a harmonic sequence or series, we
compare it to a corresponding arithmetic sequence or series
whose terms are the reciprocals of those of the harmonic
sequence.
• Find the 36th term of the harmonic series
1 1 J_ JL_
4 + 7 + 10 + 13 +' •■"
The reciprocals of the given terms form an arithmetic series,
1 + 4 + 7 + 10 + 13 + ..., whose common difference is 3. The 36th
term of the arithmetic series is
1 + (36-1)3 = 106;
the corresponding term in the harmonic series is ttt- . •
In music, vibrating strings of the same material and with
equal diameter, equal torsion, and equal tension and whose
lengths are proportional to terms in a harmonic sequence
Harmonic Tones generate harmonic tones.

Section 8.2 Finite Sequences and Series 267
The Sum of a Finite Arithmetic Series
The sum Sn of a finite arithmetic series consisting of n terms
is given by
~ n .
Sn = "J <a 1 + an) ;
that is, the sum of an arithmetic series equals half the product
of the number of terms and the sum of the first and last terms.
Why? Write the sum of the arithmetic series twice, the second time in
reverse order; then add the terms in pairs and solve for Sn :
Sn = a\ + fai + d) + fai + 2 d) + ... + [a\ + (n -1) d]
Sn = [al + (n-l)d] + [ai + (n -2)d] + [a>i + (n -S)d] +...+ a\
2Sn = [2 ax + (n- l)d] + [2 ax + (n - l)d] + [2ax+ (n - l)d] +...+ [2ax+ (n - l)d]
2Sn = n \2 a\ + (n - 1)d\ = n \a\ + a\ + (n - l)d]
sn = "2 Ial + ^1 + (^- Dd]
Since the last term of the series is an, and an = ai+(n- 1) <i,
we have
To illustrate with numerical values, find
50
i = i
the sum of all positive, even integers from 2 to 100.
Again, write the sum of the series twice - the second time in
reverse order - add the sums, and divide by 2 :
S50= 2+ 4+6 + 8+...+ 94 + 96+98+100
S50 = 100+ 98+96 +94 +...+ 8 + 6 + 4 + 2
2S50 = 102 + 102 +102 + 102 +...+ 102 + 102 + 102 + 102
#50 = Y" © + 10°) = 2550 •
• Find the sum of the series
5 7
1 + r- + r- +...+ 201.
The common difference is r-; if the number of terms is x, then
201 = 1 + (x - 1) I ; x = 301.
And the sum is
301'
-5-(1 + 201) = 30 401.

268 Chapter 8 INTRODUCTION TO SEQUENCES AND SERIES
Geometric Sequences and Series
In general terms, a finite geometric sequence is written
al> air, air2> air^> •••> air11'1,
where a\ denotes the first term, n is the number of terms, and r
is the common ratio; the geometric sequence
3* ? 1, 3, 9, 27
has the common ratio 3.
In a geometric sequence or series, the rc-th term is
an = ai rn ~1,
where r is the common ratio.
The series
4 + 8 + 16 + 32 + ...+0^
has the common ratio
The 17th term is
a17 = 4/2^-1) = 262144.
Geometric Mean
The geometric mean m of two numbers a and b is the square
root of their product,
a m i—-
— = T" J m = ya b .
m b
The geometric mean of 4 and 4096 is
V 4 • 4096 = V16 384 = 128 .
The Sum of a Finite Geometric Series
The sum Sn of a finite geometric series consisting of n terms
is given by
o i-r" ai-ran
S" = ai l-r = 1-r—' '
where a\ denotes the first term, n the number of terms, and r
the common ratio.
To determine the sum of a finite geometric series, we must
know the number of terms, the first term, and the common
ratio.

Section 8.2 Finite Sequences and Series 269
Why? If the sum of the finite geometric series is
&n = a\ + a\r + a\r2 + a\r^ + ...+a\rn~1,
then
rSn = a\r + a\r2 +a\r3 + ... + a\rn~1 + airn .
Subtracting, all terms on the right-hand side disappear except
two,
Sn(l-r) = ai(l-r");
and
sn = Qi 1-r ; r * 1.
19
• Determine ^ 4 (2l) .
i = 0
We are asked to complete the sum of a geometric series where
the first term is 4, the number of terms is 20, and the common
ratio is 2,
4-20 + 4-2^4-22 +... + 4-219.
19
xn 1-220
2,4(2') =4 = 4(220-l) = 4194300.
i = 0
• Find the sum of the geometric series
2 + 6+18+... + 39366.
The common ratio is 3.
If the number of terms is x, then
Exponential Equations: 39 366 = 2 • 3(*~*>; 19 683 = 3(*" 2>.
p. 314
(x- 1) lg 3 = lg 19683 = 9 • lg3
x = 9+1 = 10.
The sum of the series is
1-31°
1-3
= 310-1 = 59 048.
Assuming that one generation corresponds to 25 years, find
the number of ancestors - parents, grandparents, great-
grandparents, etc. - a person might have over a 600 year
period.
600 / 25 = 24 generations
2 parents + 4 grandparents + 8 great-grandparents + ...
+ (2) 224 ~ 1 great- .. .great-grandparents.
24
xn 1-224
2,2' = 2 + 4 + 8+... +224 = 2 = 2(224-l)
i = 1
= 33 554 430 ancestors.

270
Chapter 8 INTRODUCTION TO SEQUENCES AND SERIES
Legend has it that the game of chess was invented for the
amusement of a Persian shah - or an Indian maharajah,
or a Chinese emperor - who became so enthusiastic that he
wanted to reward the inventor, who desired only one grain of
wheat on the first square of the chessboard, two grains on the
second square, four on the third, and so on, doubling the
number of grains for each successive square on the 64-square
chessboard.
The shah - or maharajah, or emperor, or whatever he was -
smiled to himself at this modest wish; but the smile froze
when next morning his Keeper of the Granary told him -
with the help of his electronic calculator, no doubt - that he
would have to shell out
1-264
1 + 2 + 4 + 8 + 16 + 32 + ... +263 = 1-—— = 264-l grains,
that is,
1-2
18 446 744 073 709 551615 grains of wheat.
With about 100 grains to a cubic centimeter, the total volume of
wheat would be nearly two hundred thousand million cubic
meters, or two hundred cubic kilometers, to be loaded on two
thousand million railway wagons, which would make up a
train reaching a thousand times around the Earth.
8.3 Infinite Series
Further discussion:
pp. 355 et seq.
As more terms are added to a convergent series, its terms
become arbitrarily small. We say that the terms of a
convergent series tend to 0. The sum of such a convergent series is
referred to as the sum to infinity, and is denoted
lim S
n f
n—»°o
where "lim" stands for "limit" (Latin: limes) and indicates
"boundless approach" or "nearness".
For a series to converge its terms must tend to 0, but this is not
the only requirement for convergence; even if the terms tend
to 0, the series does not necessarily converge.
The n-th Term Test
The following theorem, often referred to as the n-th term test,
provides a simple test to determine the limit of the terms of a
sequence or series.
oo
If a series 2^ an is convergent, then
n = l
lim an = 0 .
n —><»

Section 8.3 Infinite Series
271
Why? We have
lim Sn = lim {a\ + c*2+ ... +a>n-l + an)
n —><» n —» °°
= lim (Sn_i + an) = lim Sn_i + lim an
n—>°« n—>°« n = °«
Solving for a n , we find
lim an = lim (Sn -Sn_i) = 0 .
/i —> oo n —>°o
Convergence and Sums of Selected Series
We shall demonstrate in the following examples that there are
other requirements for convergence of an infinite series than
that the terms must tend to zero.
1. Consider the series
n + 1
= lim
n —> °°
(j-2l +
(i i\ (i l
l2"3l +
^3"4I +
U"5I+- +
(1
n + 1
where
lim
n —»°o
fI__L
^71 71 + 1
= 0;
the terms tend to zero, a prerequisite for convergence.
The series has the partial sums Si, S2, S3, S4,..., S n :
Si = 1-t =
0 1 X X X 1 X
^2 = 1"2 + 2"3 = 1"3 =
^3 = 1"2 + 2"3 + 3"4= 1_4 =
_ 1 1_1 1_1 1_1_1_1_£
^4 ~ 2 + 2 ~3 +3~4 + 4~5 ~ 5"5
1
2
2
3
3
4
lim Sn = lim
n
n—>«>
n —> 00
1-
n + 1
= 1-0 = 1.
Consequently, the series is convergent and its sum is 1.
2. Consider the series
As
00
I
n = l
1 J_ J_ J_ J_
f ^~ ^. 1 «■■ ■ 1 «■■ ■ 1 «■■ ■ 1^ • • • 1 f 1^ • • •
Vrc V2 V3 V4 Vrc
lim ~7= = 0,
the terms tend to zero.

272 Chapter 8 INTRODUCTION TO SEQUENCES AND SERIES
The partial sum Sn has n terms, the smallest being 1/Vrc, so
1 A
Sn > n , .—
cinJ
= Vrc~ .
When n tends to infinity, so does yn . The series is thus
divergent; it has no sum.
3. Consider the harmonic series
oo
I
n = l
As
JL_11111 1_
rc 2 3 4 5 6 n
lim -= 0,
n -»00 ^
the terms tend to zero.
Starting with the third term, we add blocks of terms,
containing 2, 4, 8, 16, 32, 64, ... terms, etc.
The sum Sn of each block will be greater than the product of the
number of terms times the smallest term of the block:
1st term 1 = 1
2nd term 1/2 = 1/2
3rd and 4th terms 1/3 + 1/4 > 2 (1/4) = 1/2
5th to 8th terms 1/5 + 1/6 + 1/7 + 1/8 > 4 (1/8) = 1/2
9th to 16th terms 1/9 + 1/10 + ... + 1/16 > 8 (1/16) = 1/2
17th to 32nd terms 1/17 + 1/18 + ... + 1/32 > 16 (1/32) = 1/2
We can continue indefinitely to find blocks of terms whose
sum exceeds 1/2; Sn will thus tend to infinity as n does and,
consequently, the series is divergent.
4. Consider the series
oo
]T (-i)» = -1 + 1-1 + 1-1 + 1-1 + ... + (-1)^ + ....
n = l
As the limit value
lim (-1)"
n —> oo
does not exist, the terms cannot tend to zero; the series is
divergent.
It might be tempting to assume that the sum is either (- 1) or 0,
depending on the last term. However, there is no last term -
the terms continue indefinitely.
5. It is obvious by the very definition of an arithmetic series that
the magnitude of its terms becomes ever greater.
Consequently, infinite arithmetic series expand beyond bounds and
never converge.

Section 8.3 Infinite Series
273
Summarizing, we have the following facts concerning
infinite series:
o A series whose terms do not tend to zero is divergent.
o A series whose terms tend to zero may or may not be
convergent.
Euler's Constant
Although the harmonic series
oo
I
n = l
J^lllll 1_
n 2 3 4 5 6 n
diverges, there is a formula for an approximate value of the
sum of a finite number of its terms:
m
I
n = l
— & In m + y,
n
where In m is the natural logarithm of the number of terms,
and / is Euler's constant; the accuracy of the formula
increases with increasing values of m.
Chapter 22: "Power Series", Euler discovered this formula in 1731 by starting with the
pp. 765 et seq.
In I 1 + -
x J
logarithmic function
— - + - -..., or — = In
* 2x2 3x3 x
and letting x — -L, ^2, o, ... n\
X + 1
\
X J
2x2 3 xs
I • • • » ^V —- ^*9
1 , o 1 1 1,3 1
T=ln2 +—- — +...; — = In— +
1 23 '2 22-43-8
1 1 , n+1 1 1
— +...; — = In +—-- —r
n n 2n2 3 rc3
+ ...
Adding terms on both sides, we get
, 1 1
1 + — + ... + —
2 n
= In (n + 1) +
if- l M
2^1 + 4 +"-+ n2
1 fi * M
-1 + -+ +
3^ 8 +-+ „3j
+ ...
or, substituting Cn for the sum of the infinite number of terms
enclosed by brackets above,
1+- + - + ...+— = In (n + 1) +Cn.
& o n
Subtracting In n from both sides gives
1+^: + 0+.-.+-- In n = In 1 + -
2 3 n \ n)
+ C„,
lim In 1+-
= 0
and as n tends to oo, we obtain
/= lim 1 + -+- +
1 , "l
+ — - In n
n )
where y is Euler's constant.

274 Chapter 8 INTRODUCTION TO SEQUENCES AND SERIES
Thus far, no one has been able to determine whether Euler's
constant is rational or irrational; to do so would be akin to
grabbing the mathematicians' brass ring.
Euler's constant has been calculated to more than 20 000
decimals; to 26 places of decimals, we have
/= 0.577 215 664 901532 860 606 512 09....
1000
Find an approximate value of
n = l
^^ n
1000
I
n = l
— * In 1000 + 0.5772 * 7.48
n
Infinite Geometric Series
Convergence
An infinite geometric series
al + air + alr 2 + alr 3 + • • •
is convergent only if the common ratio r lies strictly between
- 1 and +1.
More specifically, denoting the common ratio r\
o If | r | < 1, the infinite series converges.
o If | r | > 1, the value of | r | n increases without limit; the
infinite series diverges.
o If | r | = 1, the terms of the series either are constant or
differ by successive sign changes; again, the infinite
series diverges.
The Sum of Infinite Geometric Series
To find the sum of an infinite geometric series, we must know
the first term of the series and the common ratio.
The sum Sn of the finite geometric series is, as we have seen,
sn = fli 1-r ; r * 1,
where a\ is the first term and r is the common ratio.
When n approaches infinity, the series is convergent only if
| r | < 1 , in which case rn will tend to zero; thus,
a!(l-r") a! (1-0)
lim Sn = lim
n —><» n —> oo
n 1 - r 1-r
that is, the sum of an infinite, convergent geometric series is
T O fll
hm Sn = .
n ->°o ± — /

Section 8.3 Infinite Series
275
Find the sum
11111
— + — + + + 4-
4 8 16 32 64
The first term of this geometric series is j and the common
ratio is tt ; the series is convergent and, consequently,
oo
I
lflY ,. Q 1/4 1
n = 0
Find the product
oo
2~n
nio
n = 0
The product may be written
1Q1 + 1/2 + 1/4+ 1/8 + 1/16 + ...
where the exponents form an infinite geometric series whose
first term is 1 and whose common ratio is 1/2; the sum of the
exponents is
__i 2
1-1/2-^
and the sought product is
102 = 100.
Express the periodic infinite decimal fraction
0.16717171...
as a fraction.
16 71 71 71
0.167 17171... - 10o + 10 000 + 1000 000 + 100 000 000 *""
The terms succeeding 16/100 form an infinite geometric
series with the first term 71/10 000 and the common ratio
1/100; thus,
«„„,„, n, 16 71/10 000 1584 + 71 331
0.16717171... = ttt +
100 1- 1/100 9900 1980 *
Achilles and the Tortoise
ZENO The Greek philosopher and mathematician Zeno of Elea pro-
(c. 490 - c. 435 B.C.) pounded that in a footrace between Achilles and a tortoise,
Achilles would lose the race - no matter how fast he runs - if
the tortoise had been given a head start - no matter how short.
Zeno argued that, although Achilles runs faster and would get
closer and closer, he would never quite catch up to the tortoise
since, while Achilles covered the distance from his starting
point, a, to that of the tortoise, b, the tortoise moved to c, and

276 Chapter 8 INTRODUCTION TO SEQUENCES AND SERIES
while Achilles dashed to c, the tortoise scuttled off to d, and so
on in intervals that became shorter and shorter but never
ceased to be produced.
It is likely that Zeno fabricated this and other anecdotes to
demonstrate mathematical shortcomings of his time, but we
are not aware of any resolution of Zeno's to this or any other of
his paradoxes.
What is the fallacy of the paradox of Achilles and the tortoise?
Using concepts established some 2500 years after Zeno, here is
the explanation why Achilles can finally catch up to and pass
the tortoise:
Although the number of time intervals is infinite, the
total amount of time is not necessarily infinite.
If the tortoise is given a head start of 3 meters and advances at
the speed of 3 m/s (cheating - pulled by a tractor) and Achilles
ambles along at 6 m/s, Achilles will catch up to the tortoise at
the end of
1111 1/2^
2 + 4 + 8+16 + - = TTi/2 = X second-
Alternatively, you could solve 6t = 3t + 3 for the time t.
Convergence Tests
While geometric infinite series are easily assessed as to
convergence and sum, infinite series in general need further
consideration.
Series of Positive Terms
We will discuss two convergence tests for series with positive
terms only - the term comparison test, and the ratio test.
Comparison Test
o A series of positive terms is convergent if the value of
each term is equal to or less than the value of the
corresponding terms of another series of positive terms
which is known to be convergent.

Section 8.3 Infinite Series
277
d'Alembert's Ratio Test
d'ALEMBERT Jean Le Rond
(1717-1783)
CAUCHY Augustin Louis
(1789-1857)
o A series of positive terms is divergent if the value of
each term is equal to or greater than the value of the
corresponding terms of another series of positive terms
which is known to be divergent.
Compare the series
(1)
1 1
1 + T T +
2 • 3 3 • 32
with the geometric series
n 3""1
+ ...
(2)
, 1 1
1 + — + —- + ... +
3 32
3n-l
+ ...
whose common ratio is l/3 and therefore converges.
Corresponding terms of (1) are less than the values of the terms
of (2):
1111 1 1
i = i;
< t;
23 3 ' 332 32 ' 713*-1 3*"1
Consequently, (1) is convergent.
Ratio Test
It is often difficult to find a series suitable for convergence
comparison. A special case of the comparison test, called
d'Alembert's ratio test, provides an easier solution when it is
applicable.
Although named after the French mathematician and
physicist d'Alembert, the test was not formally stated and
proved until after dAlembert's time, by the French
mathematician Cauchy.
oo
The series 2^ an = a\ + a<i + ... + an (assume positive terms)
n= 1
dn + i
(A) is convergent if lim < 1
n —> oo
a
n
(B) is divergent if lim -2a+l > 1#
n —> oo
a
n
Why? Consider the two cases r < 1 and r > 1.
A. r<l
Assuming that lim —^—= r and that r < 1, let P be any real
n —> oo
a
n
number such that r < P < 1; there must then exist a positive
integer S such that
a>n + l
a
n
Thus,
< P, or an +1 < Pan, for any integer n > S .
«s + l < P<*s,
as + 2 < Pas + l < p2<*s,
as + 3 < Pas + 2 < PSaSf
or, in general,
for any integer k > 0 .
<*>s + k < Pkas

278 Chapter 8 INTRODUCTION TO SEQUENCES AND SERIES
The geometric series
(1) Pas + P2as + P3as+...+Pkas + ...
is convergent since the common ratio P < 1.
Each term of the series
(2) as + 1 + as+2 + % + 3 +...+ as+k + ...
is smaller than the corresponding term of (1); by the
comparison test, (2) is convergent, and so, since adding a
finite number of terms does not affect convergence, the
original series a\ + a<i + ... + an is convergent as well.
B. r>l
If lim —^-^- = r and r > 1, there must exist a positive integer
S such that
an + 1
«n
> l,oran + 1 > an,
for any integer n > S, and we have
an * 0.
oo
Consequently, if lim —^-^- > 1, then 2^ an diverges.
n _> oo an n _ i
The ratio test is inconclusive if lim = 1 or if the limit
n _> oo n
does not exist; in such cases, we must resort to other tests for
convergence.
oo
Does the series y converge?
I
n = i
3. gy+i. (,+1)2
lim = lim
3 A 2 1 A 3 3
= 7 lim 1 + -+-T =7(l + 0 + 0) = 7
As lim " + = 7 < 1, the series converges.
n —> oo n

Section 8.3 Infinite Series
279
oo
Does the series
I
ft = 1
:A
n'
1 _
n
5)
converge?
lim = lim —
ti
n —> oo
a
n
ft —> oo
5 n + 1
6,. //1 + 1 1 A
- lim 7 - 7
ft —> oo \ /
As lim n + = — > 1, the series diverges.
ft —> oo 't
OO
Does the series
I
ft = l
v^~
converge?
p. 272
lim
ft —> oo
an + 1
a
ft
lim
ft —> oo
Vn + l
1
lim
ft —> oo
\rc + 1
lim
ft —> oo
\rc + 1
71 + 1
= Vi-o = l.
Consequently, the ratio test is inconclusive. We have already
seen, however, that the series diverges.

280 Chapter 8 INTRODUCTION TO SEQUENCES AND SERIES
Convergence of Alternating Series
While a series with continually decreasing positive terms,
approaching zero, converges only when special conditions
are met, an alternating series with terms whose absolute
values are continually decreasing towards zero will always
converge. The latter part of this statement is restated in the
following theorem.
The alternating series
oo
2^ (-l)n-ian = a1-a2 + a3-a4 + a5-a6 + ... (an > 0)
n = l
converges if
an + i < an for all n
lim an = 0
n—»°o
Why? We content ourselves with an intuitive explanation.
Consider the partial sums
Si = ai,
S2 = (ai-a2) < Si,
S2 < S% = (ai-a2 + a3) < Si,
S4 = (ai~a2+ 0,3-0,4) < S3,
S4 < S5 = (ai-a2 +0,3-0,4+ 0,5) < S3,
Sq = (ai — a2 + (23 — 04 + 0,5 — ag) < S^, etc.
Placing the values of Si, S2, S3, ... on the real number line, we
see that for each addition or subtraction of a term the values of
the partial sums, in an oscillating manner, are closing in on
a definite value SL .
0 S2 S4 S6
S5 Ss Si
S
00
Hence, the alternating infinite series converges.
Convergence, Absolute or Conditional
A series is said to be absolutely convergent if it converges
when the terms are replaced by their absolute values. If it is
convergent but diverges when its terms are replaced by their
absolute values, it is said to be conditionally convergent.
00
Absolute convergence leads to convergence, for if V \an
1
p. 276 converges then - according to the comparison criterion
00 00
j i ( I an I ± an) will converge, as will the difference V an .
1 1

Section 8.3 Infinite Series
281
The harmonic alternating series
oo
I
m ^ (-ly+ * , i i i
n=l
converges.
Replacing the terms by their absolute values, we have the
divergent harmonic series
oo
^^ I f_1 Vi + l
(-1)'
n
n = l
i 1 1 1
2 3 4
Consequently, (1) is conditionally convergent.
On the other hand, the alternating series
oo
I
() ^ 4 I 2 J ~ 4 8+16 32 + 64
n = 0
is absolutely convergent, for the series
oo
n = 0
_ 1 1 J_ J_ JL_
"4 + 8+16 + 32+64 +*"
is convergent.
Methods for testing series of positive terms for convergence
are also applicable for testing alternating series for absolute
convergence.
Approximating Sums and Remainders
The remainder of an infinite convergent series is the sum of
terms remaining after adding any chosen number of the first
terms of the series. If we add the first four terms of
oo
^T an = a\ + a<i + as + a4 + a5 + clq + ...,
we have an approximate value of the sum,
oo
X% ** a\ + a<i + as + a±,
n = l
whose error is the remainder
oo
^T an = (1% + Og + (27 + <2g + ag + aio + ....
/i = 5

282
Chapter 8 INTRODUCTION TO SEQUENCES AND SERIES
Series of Positive Terms
p. 278
Since the exact sum of a convergent geometric series can
always be determined, the remainder of such a series is also
readily calculable. The series
oo
4 V2j "4 + 8+16 + 32+64 +'"
n = 0
has the sum
1/4
1-1/2 2*
The first 10 terms give the sum
00
SKi
2
n
'i\ 1-(1/2)1°
A) 1-1/2
1023
2048
n = 0
The error is then the remainder
1
2
1023
2048 2048 *
With the exception of geometric series, it is generally
cumbersome to find the sum of a convergent infinite series of positive
terms, especially if it converges slowly. ("Slow" and "rapid"
convergence are relative notions, referring to the number of
terms needed to attain a desired accuracy.)
However, with the assistance of computers, we can easily
obtain approximate values of virtually any convergent series;
the accuracy is determined by the number of terms added.
To determine the sum
00
I
n = 1
n2Sn + 1
4ft
1(32) 22 (33) 32 (34) 42 (35)
+ — + — + :— +
42
43
44
tested for convergence in a previous example, we program a
computer to add 1000 terms, displayed with 10 decimals:
N
O
N
I
/i=l
n2Sn + 1
4«
1
2
3
4
5
2.2500000000
9.0000000000
20.3906250000
35.5781250000
53.3759765625

Section 8.3 Infinite Series
283
10
•
•
20
•
100
•
•
119
122
123
124
•
•
999
1000
146.5811004639
•
•
235.2179464192
•
•
251.9999999687
•
•
251.9999999998
251.9999999999
251.9999999999
252.0000000000
•
•
252.0000000000
252.0000000000
One must be cautious with this kind of numerical calculation,
for divergence may be so slow that it does not show for what, at
first, may seem a reasonable number of terms.
Alternating Series
Unlike convergent infinite series of positive terms, the
magnitude of the remainder of a convergent alternating
series may always be estimated at any point in the succession
of terms.
The absolute value of the remainder of the sum of an
alternating series,
oo
n= 1
is equal to or less than the value of the first deleted term.
Consequently, if the sum of the series is S, the sum of the first./
terms is Sj, and the sum of the remaining terms is Rj + i, then
\S-Sj\ = \Rj+1\<aj+1 .
Why? We have
~ j
RJ + 1 = S-Sj= X(-D" + 1«» - X(-1)n + la»
/1=1 /1=1
= (-1)/0,- + 1 + (-lV + 1aj + 2 +(-lV + 2aj + 3 + ...
= (-1)/(0/+1-0/ + 2 + 0/ + 3--)
or
l^/'+ll = aj+l -aj + 2+aj + 3-aj + 4 + aj + 5-~'
= CLj+1 -(0/+2-^/ + 3)-(^ + 4-^ + 5)----
and, since every expression in parentheses yields a positive
term,
\Rj+l\ ^ 0/ + 1 •

Chapter 8 INTRODUCTION TO SEQUENCES AND SERIES
Consider, for instance, the convergent alternating series
oo
I
n = l
(-1)/» + 1 [ I>
P )
,_1 11
2 + 3 4 +**
The absolute value of the 10th term is | —l/10 | ; consequently,
adding the first 9 terms gives an "accuracy" of only about
± 1 • 10"1, and adding the first 19 terms, about ±0.5 • 10"1.
To ensure an accuracy of ± 1-10~5 requires that the first 100 000
terms be added.
(By methods of integration, it can be shown that the sum is, in
fact, In2.)
4gs
«M
n* i.

285
8.4 The Tower of Hanoi
The game of the tower of Hanoi has been played since the latter
part of the 19th century but in all likelihood is much older. The
task is to transfer and rebuild the tower around one of the
initially empty pegs in such a way that
o only one ring be moved at a time, and
o no ring must rest on a smaller ring.
For 2 and 3 rings a minimum of, respectively, 3 and 7 moves
are required:
A, 1 1
-L 1 1
J_ _L
1 1 A,
A. 1 1
A. 1 1
-L 1 1
-L 1 J,
1 J_ i.
Ill
1 A, 1
1 A. 1
The increase from 2 to 3 rings required that the two rings on
the top be piled up twice, the second time on the moved bottom
ring. We discern the pattern
1 ring => 1 move
2 rings => 2 1 + 1 = 3 moves
3 rings => 2-3 + 1 = 7 moves.
With similar reasoning, a 4-ring tower means that 3 rings
must be built up twice, a 5-ring tower requires that 4 rings be
built up twice, etc., and every time the second edifice is erected
on the moved bottom ring, giving
4 rings => 2 • 7 + 1 = 15 moves
5 rings => 2 • 15 + 1 = 31 moves
6 rings => 2-31 + 1 = 63 moves, etc.
We have
1, o, 7, 15, ol, bo,..., An _ i, An ,
where An = 2 An _ i + 1, that is, An + 1 = 2 (2 An _ i + 1).
To find the n-th term, rewrite the sequence thus:
21-!, 22-l, 23-l, 24-l,25-l, 26-l, ..., 2»-l
To rebuild a tower of 10 rings requires
210 - 1 = 1023 moves.
+.

286
Chapter 8 INTRODUCTION TO SEQUENCES AND SERIES
8.5 The Fibonacci and Related Sequences
FIBONACCI Leonardo
(1170-c. 1250)
Mathematics of
Rabbit breeding
One of the most celebrated problems concerning sequences
appears in the Liber abaci, published in 1202 by the Italian
merchant and mathematician Leonardo di Pisa, or Fibonacci
(Figlio dei Bonacci, "Son of the Bonaccis"); the problem can be
related as follows:
A pair of newly born rabbits, male and female, were placed in a
hutch. In two months these rabbits began their breeding cycle and
produced one pair of rabbits, one male and one female. Ifie original
rabbits and their offspring continued to breed in this manner, that
is, the first pair of offspring appearing at the parental age of two
months and then a new pair every month thereafter - always one
male and one female. Ml rabbits survived the first year.
What then is the total number of pairs of rabbits at the beginning
of each month during the first year?
The original pair had their first pair of offspring at the
beginning of the third month, and another pair at the
beginning of the fourth month. At the beginning of the fourth
month, 2 pairs were fertile, resulting in 2 pairs of offspring at
the beginning of the fifth month, and so on:
Month:
Beginning of
1st
2nd
3rd
4th
5th
6th
7th
8th
9th
10th
11th
12th
Productive
0
1
1
2
3
5
8
13
21
34
55
89
Pairs of Rabbits:
Nonproductive
1
0
1
1
2
3
5
8
13
21
34
55
Total
1
1
2
3
5
8
13
21
34
55
89
144

Section 8.5 The Fibonacci and Related Sequences
287
Fibonacci Numbers
LUCAS
Francois Edouard Anatole
(1841 -1891)
SIMSON Robert
(1687 -1768)
Golden Number
Lucas Sequence
A look at the resulting sequence,
1,1,2,3,5,8,13,21,34, 55,89,144, 233,377,...,
reveals that, from the third term onward, every successive
term is the sum of the two immediately preceding terms; that
is, Fibonacci numbers satisfy the recursion formula
Fn + Fn + 1 = Fn + 2
*l= 1; F2 = 1
We have no evidence that Fibonacci further explored the
sequence; nor did his name become attached to it until the 19th
century - to the best of our knowledge, in a paper by the French
mathematician Edouard Lucas in a publication devoted to
recreational mathematics. (Lucas is otherwise best known for
the fact that, in 1876, he discovered that the 39-digit Mersenne
number M427 = 2127 - 1 is a prime.)
In 1753, Robert Simson of the University of Glasgow showed
that the ratio of one Fibonacci number to the one preceding it,
1 2 3 5 8 13
1' 1' 2' 3' 5' 8 '■■■•
draws progressively nearer, alternately from above and from
below, to the golden number O,
O =i(V5 + l)= 1 +
= 1.618 03....
1 +
1 +
1 +
1 +...
Ascending integer powers of the golden number yield the
following sequence:
<D = i(V5+l),
02=f(V5 + 3),
03 = f(2V5 + 4),
04 = f(3V5 + 7),
<D5 = i(5V5 + ll),
06 = f(8V5 + 18),
<D7 = i(13V5 + 29),
<D8 = i(2lV5 + 47),
09 = |(34V5 + 76)...,
where the coefficients of the irrational \5 terms form the
Fibonacci sequence, and those of the rational terms form the
Lucas sequence
1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199,...,
whose terms satisfy the recursion formula
Ln + Ln + 1 = Ln + 2
Lx = 1; L2 = 3
That is, like the Fibonacci sequence, every term of the Lucas
sequence is the sum of the two immediately preceding ones.

288
Chapter 8 INTRODUCTION TO SEQUENCES AND SERIES
Binet Formula
BINET Jacques-Phillippe-Marie
(1786 -1855)
Pell Number Sequence
PELL John
(1610-1685)
The general terms of the Fibonacci and Lucas sequences are
given by the Binet formulas for Fn and Ln :
Fn =
V5
1 + V5Y (i - V^
Ln =
l + VsY (i - Js
The Pell number sequence
Pn: 1, 2, 5, 12, 29, 70, 169,...,
named after the English mathematician John Pell, and
Qn: 1, 3, 7, 17, 41, 99, 239,...
satisfy the recursion formulas
Pn + 2Pn + 1 = Pn + 2
Pi= 1; P2 = 2
and the Binet formulas
Qn + 2 • Qn + 1 = Qn + 2
Qi = l; «8 = 3
p„ = ^[(i+^)Mi-^n
2V2
Q„ = |[(i+^)%(i-^)"]
Fibonacci numbers and related number sequences appear as
natural phenomena, such as the shape of snail shells and the
heads of sunflowers, and in phyllotaxy. They have also
proved relevant and useful in branches of mathematics, for
instance, the theory of equations, and in the study of genetics,
in electronics, and in data handling of statistics.
The Fibonacci Association was founded in 1962 in California
to further interest in Fibonacci numbers and related topics and
has published, since 1963, The Fibonacci Quarterly.

289
8.6 Figurate Numbers
Triangular Numbers
Arrangements of dots to represent numbers as geometrical
figures, found as far back as Stone Age rock carvings, were of
special importance to the Pythagoreans (c. 6th century B.C.),
who imparted numbers with specific characteristics and
personalities and believed that everything could be explained
by numbers. Mystic or divine attributes of numbers were
prevalent also among the Babylonians, ancient Maya, and
most other ancient cultures.
The Pythagoreans demonstrated many of the arithmetic
features of figurate numbers.
Beyond serving as number games, figurate numbers lead to
interesting and useful progressions and series of numbers,
and they give us ways to visualize and geometrize relations
between various sorts of numbers.
Triangular numbers are the natural numbers which can be
drawn as dots and arranged in triangular shape: 1, 3, 6, 10, 15,
21, 28, 36, 45, 55, 03, etc.
1
2
4
3
5
7 8
TerpaKTba
6
9
11 12 13
10
14
16 17 18 19
15
20
22 23 24 25 26
21
27
29 30 31 32 33 34
28
35
36
10
15
21
Of all numbers, 10 was held in greatest reverence by the
Pythagoreans; the sum 1 + 2 + 3 + 4 = 10 was named tetraktys,
"the holy fourfoldness", representing the four elements: fire,
water, air, and earth.
By writing the numbers in rows of increasing length, as
shown in the figure to the left, we see the progression of
triangular numbers.
The n-th triangular number Tn is given by the sum of an
arithmetic progression of natural numbers,
Tn = 1 + 2 + 3 + . ..+n,
which may be written
2 Tn = n (n + 1)
n (n + 1)
Tn =
the figure shows why:
o
o
o
o
o
o
o
o
o
•
o
o
o
•
•
o
o
•
•
•
o
•
•
•
•
2 Tk = 5 • 6

290
Chapter 8 INTRODUCTION TO SEQUENCES AND SERIES
Square Numbers
The sum of two consecutive triangular numbers is the sum of
consecutive odd integers starting at 1,
Tn + Tn + 1 = 1 + 3 + 5 + ...+(2/z + l);
and here is why:
o
oo
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
•
•
•
•
1
•
•
•
3
•
•
5
•
7
9
It has been proved that every integer is either a triangular
number or the sum of two or three triangular numbers.
Square numbers, 1, 4, 9,16, 25, 36, 49, 64, 81,100,121, 144, etc.,
are figurate in this way:
16
25
36
The sum of two consecutive triangular numbers is always a
square number:
o
o
o
o
o
o
o
o
o
o
o
•
o
o
o
o
•
•
o
o
o
•
•
•
o
o
•
•
•
•
o
Tn + Tn _ ! = n2
Since the sum of two consecutive triangular numbers is equal
to the sum of consecutive odd integers starting at 1, we also
have
n2 = Tn + Tn_1 = 1 + 3 + 5 + ... + (2/2-1).
The main diagonal of the multiplication table is composed of
square numbers:
1
2
3
4
5
6
7
8
9
10
11
12
2
4
6
8
10
12
14
16
18
20
22
24
3
6
9
12
15
18
21
24
27
30
33
36
4
8
12
16
20
24
28
32
36
40
44
48
5
10
15
20
25
30
35
40
45
50
55
60
6
12
18
24
30
36
42
48
54
60
66
72
7
14
21
28
35
42
49
56
63
70
77
84
8
16
24
32
40
48
56
64
72
80
88
96
9 10 11 12
18 20 22 24
27 30 33 36
36 40 44 48
45 50 55 60
54 60 66 72
63 70 77 84
72
81
90
99
80 88 96
90
100
110
108 120
99 108
110 120
121
132
132
144

Section 8.6 Figurate Numbers
291
LAGRANGE Joseph Louis
(1736-1813)
Gnomons
>f¥
Lagrange proved in 1770 that every natural number is the sum
of no more than four squares.
Gnomons are the geometric representations of odd numbers as
dots on equally long legs of a right angle. The name refers to
the angle's likeness to the Babylonian sundial, the gnomon.
13 5 7 9 11
By adding gnomons, the Pythagoreans built larger squares
from which they deduced many interesting connections
between numbers.
A square thus formed demonstrates the relationships
1 + 3 = 22,
1 + 3 + 5 = 32,
1 + 3 + 5 + 7 = 42,
1 + 3 + 5 + 7 + 9 = 52,
1 + 3 + 5 + 7 + 9+11 = 62,
1 + 3 + 5 + 7 + 9+11 + 13 = 72,
Oblong Numbers
and generally,
1 + 3 + 5 + 7+ ... + 2n-l = n2 .
Oblong numbers, 2, 6,12, 20,30,42,56, 72,90,110,132, etc., have
a number of dots that can be placed in a rectangular pattern:
Pentagonal Numbers
P5 = T5 + 2T4
12
20
30
These numbers lack particular interest in mathematics, but
justify attention because "square - oblong" is one of the ten
"cosmic opposites" in the Pythagorean doctrine ofopposites.
The doubling of a triangular number yields an oblong
number,
On = 2 Tn = n (n + 1).
Pentagonal numbers, 1,5,12,22,35,51, 70,92,117,145, etc., are
figurate in this way:
.&
&€£
12
22
Pentagonal numbers can be generated by triangular ones, as
seen here to the left, yielding the general formula
Pn = 1 + 4 + 7+... + (3^-2) = Tn +2Tn_i
= -rn (n + 1) + n (n - 1) = ^71 (3 n - 1) .

292
Chapter 8 INTRODUCTION TO SEQUENCES AND SERIES
We now have the relations:
natural numbers n,
triangular numbers Tn,
square numbers n2,
pentagonal numbers Pn,
and may continue:
hexagonal numbers Hexn,
heptagonal numbers Hepn,
n
T
n
P
n
2
n
= Tn+ (-l)TB_i
= Tn+ o-rn_i
= Tn+ i-r„_i
= rn+ 2-rn_i
Hexn = Tn +
HePn = Tn +
Octn = Tn +
octagonal numbers Octn,
and the polygonal number with p sides, n in each side, is
r„ + (p-3)-rn_i.
3-r„_i
4-r„_i
5-r„_i
Three Dimensions
Cubic Numbers
We may extend the scope of figurate numbers to three-
dimensional space: tetrahedral numbers, pyramidal
numbers, cubic numbers, etc.
To build cubic numbers, 1, 8, 27, 64, 125, 216, etc., we
successively stack n ■ n squares n high.
A^r*
8
27
64
Tetrahedral Numbers
In the succession of positive odd numbers,
X, O, O, /, c/j -1--1-9 -*"*^9 J-Oj J- I , J-«^j • . •,
we have
1 = 13,
3 + 5 = 23,
7 + 9+11 = 33,
13 + 15 + 17 + 19 = 43,
etc.,
the general formula being
[n (n - 1) + 1] + [n (n - 1) + 3] + ... + [n (n - 1) + 2n
-1] =
n*.
To build tetrahedral numbers, 1, 4, 10, 20, 35, 56, etc., we
successively stack triangular numbers, 1, 3, 6, 10, 15, ..., one at
a time.
4>
10
20

Section 8.6 Figurate Numbers
293
1 4 10 20 35 56
/////
12 3 4 ^ fi
// /// The general formula for a tetrahedral number of n layers is
2 m 6 8 10
//L/ / Thn = n- 1 + (/2-1)-2 + (72-2)-3 + (72-3)-4+... + 2(/2-1) + 1-72
3 6 00 12
/ // 1
4 8 12 = - 72 (/2 + 1) (/2 + 2) .
// 6
5 10
/ By adding diagonals of the multiplication table, as shown here
to the left, tetrahedral numbers can easily be found.
6
Square Pyramidal Numbers Similarly, to build square pyramidal numbers, 1, 5, 14, 30, 55,
91, etc., we successively stack the square numbers, 1, 4, 9, 16,
25,36, etc., one at a time.
A
14 30
To obtain the n-layer square pyramid, Pyramn, add the first n
squares,
Pyramn = l2 + 22 + 32 +...+/22,
whose sum can be found by a formula derived from the general
formula for a tetrahedral number: The sum of two consecutive
triangular numbers is a square, so the sum of two consecutive
tetrahedral numbers is a square pyramidal number,
Pyramn = — n (n + 1) (72 + 2) + -(/2-1)/2 (/2 + 1)
b b
= - /2 (/2 + 1) (2 /2 + 1).
D
Four Dimensions
Figurate numbers may be formed in four or more dimensions,
but these shapes are extremely hard, if not impossible, to
visualize.
Sums of Cubes
Adding on cubes, we have
13+ 23+ 33+ ...+/23.
Writing
13 = 1(1)
23 = 2 (1 + 2 + 1)
33 = 3(1 + 2 + 3 + 2 + 1)
43 = 4(1 + 2 + 3 + 4 + 3 + 2+1)
53 = 5(1 + 2 + 3 + 4 + 5 + 4 + 3 + 2 + 1)
n3 = n [1 + 2 + 3 + 4 + 5+...+/2 + (/2-1)+... +4 + 3 + 2 + 1],
the sum total of the right-hand column can be expressed as
(1 + 2 + 3+...+/2)(1 + 2 + 3+...+/2).

294 Chapter 8 INTRODUCTION TO SEQUENCES AND SERIES
Hence, the sum of n cubes
l3 + 23 + 33 + ...+rc3
is equal to the square of the /i-th triangular number,
-n (n + 1)
= ±n2(n + 1)2
4
"Supertetrahedral Numbers"
By piling up tetrahedral numbers, 1, 4, 10, 20, 35, 56, etc., we
make four-dimensional numbers:
1 = 1,
1 + 4 = 5,
1 + 4 + 10 = 15,
1 + 4 + 10 + 20 = 35,
1 + 4 + 10 + 20 + 35 = 70,
etc.
Adding on Dimensions
Extending from one dimension (1-D) to five (5-D), we have
1-D the /i-th counting number:
1 + 1 + 1 + ... + 1 =rc;
2-D triangular numbers:
1 + 2 + 3 + . ..+n = — /i (/i + 1);
3-D tetrahedral numbers:
1 + 3 + 6 + ...+— n(n + 1) = — /i (/i + 1) (/i + 2);
Z b
and we can go on to
4-D 1 + 4+10 + ...+- /i(/i + l)(/i + 2) =7-7/1(/1 + l)(/i + 2)(/i + 3)
d 24
and on,
5-D 1 + 5 + 15 + ... + — n (n + 1) (/i + 2) (n + 3) = ttt n (n + 1) (n + 2) (/i + 3) (n + 4)
2A 120
and on ... and there will always be another dimension.
Difficulty imagining dimensions of higher order places us in
safe and illustrious company:
It is often hztpfuC to thinf^ of the four coordinates of an event as
specifying its position in a four-dimensional space called space-time. It is
impossible to imagine a four-dimensional space, I personally find it hard
enough to visualize three-dimensional space1.
Stephen W. Hawking, A Brief History of Time (1988)

295
Chapter
9
THEORY OF EQUATIONS
Page
9.01 History 297
9.02 Groundwork 302
9.1 Linear Equations 307
9.2 Equations with Absolute Values 308
9.3 Quadratic Equations 309
9.4 Inequalities 312
9.5 Root, Exponential, and Logarithmic Equations 313
9.6 Cubic Equations 316
9.7 Quartic Equations 320
9.8 Systems of Equations 325
9.9 Diophantine Equations 330

296
^^^ -■ ■ a
Frontispiece of Thesaurus Mathematum Reservatus per Algebram
Novam (Passau, 1646) by Johann de Luneschlos, physician of great
and varied learning who in 1649 became professor of mathematics and
physics at Heidelberg.

297
9.01 History
Regula Falsi
PYTHAGORAS
(c. 582 - c. 507 B.C.)
EVCLEIDIS
(c. 330 - c. 275 B.C).
i
«—a —*
b
b
i
a
i
The Rhind Papyrus - dating from around 1650 B.C. but
probably based on a document 200 years older - contains a problem
reading:
ft quantity and its 1/7 part become IS. What is the quantity?
The problem is solved in the Egyptian manner of regula falsi]
that is, one assumes a - probably wrong - solution and makes
the following calculation:
Assume that the answer is 7,
a seventh part is then 1,
making a total of 8.
This is obviously not the right answer, and the assumed
answer 7 must be multiplied as many times as 8 must be
multiplied to give 19; that is, 8
Today we would write
19
19
-s- = 19 and 7 • -=- = 16tt .
x
x + - = 19
8 *"' 8
„ 19 ,^5
, = 7--=16-
8
The Ahmes papyrus and other ancient Egyptian scrolls are
mainly concerned with problems leading to first-degree
equations, but also describe some second-degree equations
relating to land surveying. Babylonian clay tablets from the
time of the Hammurabi dynasty (about 1800 - 1600 B.C.) deal
with quadratic equations and their solutions by the method of
"completing the square", and also describe numerical
methods of solving all quadratic equations and some simpler
forms of cubic equations.
Much of the knowledge built up by the old civilizations of
Egypt and Babylonia was passed down to the Greeks, who, in
turn, gave mathematics scientific form.
Between about 540 and 250 B.C., the ancient Greeks, represented
by Pythagoras, his followers the Pythagoreans, and Euclid,
gave strict geometric proofs to algebraic problems, using lines
and areas for numbers and products:
(a + b) (a + b) = a2 + a b + a b + b2 = a2 + 2 a b + b2
(a-6) (a-6)= a2-ab-ab + b2 = a2-2ab + b2
(a + b) (a - b) = a2 + ab-ab-b2 = a2- b2.
1
b
t
-*
^^^^^^i
«— a
b
:¾¾¾¾¾
:¾¾¾¾¾
t
a
i
i
b
t
a
t
^—a
:¾¾¾%
:¾¾¾¾
b
t
=> a
1
k— a
b \
+
b
t
a2 + 2a b + b2
a2-2ab + b2
a (a - b) + b (a - b) =
a2- b2

298 Chapter 9 THEORY OF EQUATIONS
To the Greeks, x2 + 2x = 8 meant a combination of a square
whose side was x length units and two rectangles of side
lengths x and 1 length units; to develop this figure into a full
square, they added a one-by-one small square.
X
X x
X
1
This form of reasoning corresponds to our method of cora-
pleting the square,
x2 + 2x = 8
jc2 + 2jc + 12 = 8 + 1
(x + 1)2 = 9
x + 1 =±3;jci = 2, X2 = -4
but with the significant difference that the Greeks would have
regarded -4 as an impossible solution, as no geometric
distance can have a negative length.
The Greek system of numeric notation, using letters of their
alphabet sometimes built several stories high, and littered with
accents, apostrophes, and other diacritic signs, may explain
why they turned to geometric interpretations instead of
arithmetic calculations.
The Greeks had considerable difficulty in solving cubic
equations since their practice of treating algebraic problems
as problems of geometry led to complicated three-dimensional
constructions.
Greek algebra did not separate from geometric methods until
around A.D. 250. This algebra, as presented by Diophantus of
Alexandria, resembles Babylonian algebra. Unlike
Babylonian mathematicians, however, who usually gave only
approximate solutions, Diophantus gave exact solutions of his
equations. He introduced symbols for frequently recurring
quantities and operations, and to denote powers of variables.
This syncopated style contrasts with the older rhetorical style
- generally used by other Greek mathematicians and by the
Arabs - which employs no mathematical symbols.
In ancient India, algebra was influenced by the Babylonian
and Greek schools of mathematics. The Hindu Brahmagupta
- around A.D. 630 - had a clear understanding of negative
numbers and of the concept of zero; he gave solutions to
second-degree equations, and outlined general methods for
solving equations containing several variables.
Around 825, in Baghdad, al-Khowarizmi composed the Hisab
al-jabr w'al-musqabalah ("Science of the Transposition and
Cancellation"), whose title is the source of the term algebra;
this monograph greatly influenced the introduction of
algebraic equations in Europe.
DIOPHANTOS
(c. AD. 250)
Greek, synkope, "cutting short"
Greek, rhetor, "speaker", "orator"
BRAHMAGUPTA
(c. 630)
al-KHOWARIZMI
(9th century)

Section 9.01 History
299
BHASHKARA
(1114-c. 1185)
FIBONACCI Leonardo
Leonardo of Pisa
(c. 1170-c. 1250)
GWTwffo iuoolff*
Ok [<®tun efdnpdo tor Cofe
JDutc(>
mirljatl SuM
SttJK{mgf|jrrgmprfi:tr:ii
CMrflctt/ btirc$ 2tttf<uibrnm
Xutcmy«tvnf<mimjat
M * J.
HltRONYMI
CAR DAN I PRjt.
STAT4T IS5IM1 MATHE,
mathici Philofopht, ac Medici *\Jf>.
ART1S MAGNAE,SIVE
DEREOVLISALGEBRAICIS LIBVNiu
jiiic(t«t«u5 0j>«fijdcArithiTvctica,^iio<l orvsptR.
IECT VM.uxfmpfit.efl inoicicne Dectmu*.
CARDANO Girolamo
(1601-1576)
VlfeTE Francois
(1540 -1603)
DESCARTES Rene
(1596 -1650)
GAUSS Carl Friedrich
(1777-1855)
In India, around 1150, Bhashkara, using letters to represent
unknown quantities, wrote the Lilavati ("The Beautiful") and
the Vija-ganita ("Seed Counting"), which suggested that
positive numbers have two square roots and negative numbers
no roots - or, with present-day terminology, no real roots.
Arabic, Persian, and Hindu achievements in algebra were
funneled to Europe - especially to Italy - in the 13th and 14th
centuries. An important link in this process was Liber abaci
("Book of the Abacus") of 1202 by the Italian mathematician
and merchant Leonardo Fibonacci, also known as Leonardo
of Pisa.
Influenced by al-Khowarizmi and later Islamic writers who
called the unknown shai, Arabic for "thing", Latin texts used
res and those in Italian used cosa ("thing"). In Italy, algebra
became known as Varte delta cosa, in England as cossike arte
or the rule ofcoss, and in Germany, die Coss.
In 1545, Cardano published his work
Artis magnae sive de regulis algebraicis,
generally known as Ars magna - which included solutions of
the cubic and the quartic equations as well as other
mathematical discoveries. The title Artis magnae ... is not to be
viewed as reference to the grandeur of his work but simply to
algebra itself - in that era often referred to as ars magna as
opposed to the lesser art of arithmetic.
No single publication has promoted interest in algebra like
Cardano's Ars magna, which, however, provides very boring
reading to a present-day peruser by consistently devoting
pages of verbose rhetoric to a solution which would be much
better served by a few symbolic equations. With the untiring
industry of an organ grinder, Cardano monotonously
reiterates the same solution for a dozen or more near-identical
problems, where just one would do.
Francois Viete, "the father of modern algebraic notation"
described the connection between positive roots - the only roots
taken note of in his time - and the coefficients of the powers of
the unknown quantities. His In artem analyticem isagoge
(1591; "Introduction to the Art of Analysis") greatly resembles
modern texts in elementary algebra.
We owe to Descartes, however, the present usage of denoting
unknowns by the last letters (x,y,z) of the alphabet, and
known quantities by the first (a, b, c). Descartes was one of the
main inventors of analytic geometry, the method for
analyzing geometric problems algebraically, which has contributed
greatly to the development of both geometry and algebra.
The fundamental theorem of algebra - which states that every
polynomial of degree n > 1, with real or complex coefficients,
has n real or complex roots - was proved by the German
mathematician Carl Friedrich Gauss in 1797. The proof was
published in 1799.

300
DEMONSTRATE NOVA
THEOREMATIS
OMNEM FVNCTIONEM ALGEBRAICAM
RATIONALEM INTEGRAM
VNIVS VARIABILIS
IN FACTORES REALES PRIMI VEL SECVNDI GRADVS
RESOLVI POSSE
AVCTORE
CAROLO FRIDERICO GAVSS.
HELMSTADII
APVD C. G. f LKCUUSSN.
»799.
EULER Leonhard
(1707 -1783)
BRING Edvard Samuel
(1736 -1798)
LAGRANGE Joseph Louis
(1736 -1813)
RUFFINI Paola
(1765 -1822)
ABEL Niels Henrik
(1802 -1829)
Gauss returned to the theorem many times, and published his
fourth and last proof in 1849, in which he extended the
coefficients of the unknown quantities to include complex numbers.
The fact that the general solution of the quartic equation
depended on that of a resolvent cubic equation inspired
Leonhard Euler, in about 1750, to attempt the solution of the
general quintic equation by reducing it to the solution of a
related quartic equation. The attempt failed, which led to a
general belief that the task might not be possible.
Edvard Samuel Bring, a Swedish university professor of
history, in 1786 managed to reduce the general quintic
equation to the trinomial x5 +px + q = 0 without the quartic and
cubic terms, but did not carry the attempt for a solution further.
Lagrange, the French mathematician and physicist, had, at
about the same time, already shown that the solution of the
general quintic equation depends on the solution of a sixtic
equation, "Lagrange's resolvent sixtic".
The first serious attempt to prove that a quintic equation could
not be solved by a finite number of algebraic operations
was made in 1803 - 1813 by the Italian mathematician and
physician Paola Ruffini, in communications to a
mathematical journal (Delia insolubilita delle equazioni algebraiche
generali di grado superiore al quarto).
The actual proof that the roots of the general equation of
fifth degree or higher cannot be expressed in terms of radicals
was given by the Norwegian mathematician Niels Henrik
Abel. At its first appearance, in 1824, in an unpretentious
pamphlet published at Abel's own expense, the proof still had
some inaccuracy. The correct proof was published in 1826 in
the very first volume of Journal fiir die reine und angewandte
Mathematik ("Journal for Pure and Applied Mathematics"),
often referred to as Crelle's Journal after its founder, the

Section 9.01 History
301
CRELLE August Leopold
(1780 -1855)
GALOIS fivariste
(1811 -1832)
Galois Theory, Group Theory
HILBERT David
(1862 -1943)
NOETHER Emmy (Amalie)
(1882 -1935)
GREGORIE James
(1638 -1675)
NEWTON Isaac
(1642 -1727)
WIENER Norbert
(1894 -1964)
TURING AlanMathison
(1912 -1954)
von NEUMANN John (Johann)
(1903 -1957)
WOLFRAM Stephen
(b. 1959)
German engineer and mathematician August Leopold Crelle,
whose recognition of mathematical prowess contributed
greatly to the advancement of mathematics. A comparable
work on algebraic equations was left by the Frenchman
Evariste Galois in a scientific testament which, when
deciphered, was found to give the criteria that an algebraic
equation must satisfy in order to be solvable by radicals, a branch
of mathematics now known as Galois - or group - theory.
Even if general polynomial equations of fifth degree and
beyond cannot be solved by the extraction of roots, they do have
solutions - according to the fundamental theorem of algebra.
A monumental personage in all main branches of modern
mathematics is the German mathematician David Hilbert,
who in 1895 was appointed to a professorship of mathematics at
Gottingen, which, since Gauss's days, was a leading
university in mathematical research and produced many
mathematicians of great inspirational leadership.
One of the many eminent figures that Hilbert had attracted to
Gottingen was the German mathematician Emmy Noether,
who, after the Nazi takeover in Germany, settled in the United
States in 1933. Noether made important contributions to our
understanding of properties of algebraic operations; among
other things, she developed the theory of non-commutative
algebras, that is, algebras where the order in which terms are
multiplied affects the answer. She is recognized as one of the
most creative abstract algebraists of the 20th century. The
Noether legacy extends beyond her own published works; her
influence is evident in the accomplishments of many
contemporary mathematicians for whom she was a great
inspiration and an unflagging teacher and collaborator.
High-speed electronic computers today allow us to find
approximate values of roots of polynomial equations of any
degree by numerical analysis. Instead of exact analytical
methods, numerical analysis uses "number-crunching"
techniques - sometimes held in contempt by mathematical
purists, who do not even recognize numerical analysis as
mathematics, although some of its techniques go back to
methods of interpolation conceived in the 17th century by
James Gregorie, Scottish mathematician and astronomer, and
by Isaac Newton. The term "numerical analysis" has been in
use only since 1947, when the Institute of Numerical Analysis
was founded in California.
Prominent figures in the design and use of electronic
computers were the American mathematician Norbert Wiener, the
English mathematician Alan Turing (celebrated for
breaking the German code Enigma during World War II), and the
Hungarian-born John von Neumann, considered one of the
most all-round and creative mathematicians of the 20th
century.
Due to work by Stephen Wolfram and others, easily managed
computer software is now available that can provide answers
in surds, n, and the base of the natural logarithm, e, making
those mathematicians who stay away from computer
assistance an endangered species.

302 Chapter 9 THEORY OF EQUATIONS
9.02 Groundwork
Types of Equations
In algebraic equations, variables appear as terms with
coefficients that are subject only to the fundamental mathematical
operations of addition, subtraction, multiplication, and
division.
Algebraic equations may also have terms with non-algebraic
coefficients; an equation with a term 2nx2, for instance, is
algebraic in x. A trigonometric equation of the type
4 sin2 x - sin x — 2 = 0,
while not algebraic in x, may quite well be regarded as an
algebraic equation in sin x, and be solved by algebraic methods.
Root equations, which feature terms in which the variable
appears in the form of radicands, also belong under the title of
algebraic equations.
Transcendental equations - that is, equations that are not
algebraic - are concerned with relationships between non-
algebraic numbers and quantities, but often have algebraic
coefficients.
There are three kinds of transcendental equations:
exponential, logarithmic, and trigonometric. To solve such equations,
graphic and numerical approximation methods are often
required.
In exponential equations the variables generally appear in the
exponents of power terms, but may also be found in bases and
coefficients. Exponential equations are generally solvable by
algebraic methods when the variable is part of an exponent.
Since logarithms are a form of exponent, there is no clear
distinction between exponential and logarithmic equations, or
between the methods of solving them.
Trigonometric equations containing trigonometric terms
with algebraic coefficients can often be solved as algebraic
equations.
Trigonometric equations have developed into a convenient
means of solving complicated algebraic equations, e.g., the
casus irreducibilis of the general cubic equation.
Terminology
Latin, sequare, "to make equal" An equation is a statement which shows, in mathematical
symbols, that two mathematical expressions are equal:
1+jc = 3-jc2
The numerical values of variables which make the equation
Roots, Solutions true are known as the roots or solutions of the equation; they
Satisfy an Equation are said to satisfy the equation: x + 3 = 5 is satisfied by x = 2.
Equations are basically of two types: identities and
conditional equations.

Section 9.02 Groundwork
303
Identities
Conditional Equations
An identity is a statement of equality that holds true for all
values of the variables in a universal set; to underline this,
the identity notation = may be employed:
2 n
(rx)
X
y
1 = (y-l)(y + l)
A conditional equation, on the other hand, is true only for
certain values of the variables. Thus, the equations
x2-l = 0; y + 3 = 5
are satisfied only by x = 1 or x = -1, and by y = 2.
Factoring
Factor
Common Factor
Common Divisor
Prime Polynomial
Prime Factor
Suddenly Christopher 9(0 bin began to tell To oh about some of the
things: Teople called "Kings and Queens and something called factors,
and a place called "Europe, and an island in the middle of the sea where no
ships came, and how to maf^e a Suction Tump (if you want to), and
when Knights were Knighted, and what comes from 'Brazil.
A. A. Milne: The House at Pooh Corner(1928). Illustrated by Ernest H.Shepard.
A factor is one of two or more quantities that when multiplied
together yield a given product. Factoring is the process of
breaking down polynomial expressions into factors, an
essential part of the solving of polynomial equations.
If a polynomial Q is a factor in every polynomial Pi ... P^, then
Q is considered a common factor, or a common divisor.
A polynomial with integer coefficients that cannot be further
factored to lesser polynomials with integer factor coefficients
is a prime polynomial or a prime factor; for instance, x + 1
and x2 + x + 1 are prime polynomials.
The Remainder Theorem
o If a polynomial in an unknown quantity x is divided by a
first-degree expression in the same variable, x - c, where c
may be any real or complex number, the remainder to be
expected will be equal to the sum obtained when the
numerical value of c is inserted for x in the polynomial.
Dividing the polynomial (jc3 - Sx2 + 6x - 3) by (x - 2) will
give a quotient (x2 - x + 4) and the remainder 5:
23-3-22 + 62-3 = 5.
x2-x + 4
X ~~ £j \X
X
3 x2 + 6 x - 3
2x2
- X2
- X2
+ 6jc
+ 2 x
4x -
4 x -
3
8
5

304 Chapter 9 THEORY OF EQUATIONS
The Factor Theorem
o If the division of a polynomial by (x - c) results in a
remainder of zero, (x - c) is a factor of the polynomial.
BEZOUT Etienne The remainder theorem and the factor theorem were suggested
(1730 -1783) by the French mathematician Etienne Bezout.
Factoring Guidelines
The factoring of a polynomial is complete when it has been
turned into a product of prime factors.
There are no universal rules for how to achieve a complete
factoring of a polynomial, but the following suggestions may
be useful.
1. If all terms have a factor in common, pull it out of the
polynomial:
x^y — 4 x2y2 + 4 x y3
= xy(x2-4xy + 4 y2)
5x6y2-5y2 = 5y2(x6-l)
= 5y2fe3 + 1) (x - 1) (x2 +x + 1)
2. Try the following formulas for breaking up the
polynomial:
x2±2xy + y2 = (x±y)2
x2—y2 = (x—y)(x+y)
jc3 ± y3 = (x ± y) (x2 + x y + y2)
x^±Sx2y + Sxy2±y 3 = (x±y)^
x2 + y2 + z2 + 2xy ±2yz±2zx = (x+y±z)2
3. Use the factor theorem to find first-degree factors; if the
coefficients of a polynomial are all integers, consider
integer factors of the constant term including 1.
In the polynomial (x2 - 2 x - 15), the values to be tried are
±1, ±3, ± 5, ± 15 .
+ 1 ^ 12
-1
+ 3
-3
+ 5
-5
+ 15
-15
(-1)2
32
(-3)2
52
(-5)2
152
(- 15)2
-21
+ 21
-2-3
+ 2-3
-2-5
+ 2-5
-2 15
+ 2-15
-15
-15
-15
-15
-15
-15
-15
-15
=
=
=
==
=
=
=
3^T
-16 *
-12 *
-12 *
0
0
20 *
180 *
240 *
0
0
0
0
0
0
*
*
Hence, x2 - 2 x - 15 = (x - 5) (x + 3).
Analyzing the polynomial
jc4-5jc3 + 5jc2 + 5jc-6
in the same manner we find that the roots are -1, +1, +2,
and +3; and the polynomial is factored as
(x + 1) (x - 1) (x - 2) (x - 3) .

Section 9.02 Groundwork
305
4. Group terms for "factoring by parts":
3xy-2x-12y + 8
= (Sxy-12y)-(2x-8)
x2 —y2 + 2y - 1
= x2-(y2-2y + l)
= x2-(y-l)2
= (x+y- l)(x-y + 1)
= (3y-2)(x-4)
5. Add and subtract equal terms to provide groups that are
more easily factored:
4x2-12xy + 8y2
(4x2 - 12 xy + 9y2)-y2
(2x-Sy)2-y2
K2x-3y)+y] [(2x-Sy)-y]
4(x-y) (x-2y)
x4 - 12 x2 + 144
= (x4 + 24 x2 + 144)- 36 x2
= (x2 + 12)2-(6x)2
= (jc2 + 6 x + 12) Oc2 - 6 x + 12)
= (jc + 3-iV3)(:c + 3 + iV3)
• (x - 3 - i V3) (x - 3 + i V3)
Factoring Frustration
More intricate problems of factoring are only rarely
encountered in practice. Yet many curricula call for seemingly
endless drills of factoring. Instead of nurturing an interest in
or at least some curiosity about mathematics, such "education"
- bordering on harassment - could block the path to more
exciting mathematical challenges.
... and then, as Took seemed disappointed, he added quicf^Cy, "6ut it's
grander than factors."
A. A. Milne, The House at Pooh Corner (1928)
Root/Coefficient Relationships
An algebraic polynomial equation is said to be in general
form if terms of all degrees below the highest degree are
present; to achieve this, it may be necessary to complete the
equation with additional terms with zero coefficients.
To bring the equation into standard form, terms in the right
member are transposed to the left side of the equality sign,
where all terms are arranged in order of descending degrees:
anxn + an_1xn~1 + ... a1:c + a0 =0
The equation is in monic form when all terms are divided by
the coefficient of the highest-degree term:
xn +pn_lxn~l + ...p^x + q =0
The fundamental theorem of algebra states that every
polynomial of degree n > 1, with real or complex coefficients, has n
real or complex roots. Hence, every polynomial equation of
degree n can be written in the form
(x-ri)(x-r2)(x-rs)...(x-rn) = 0,
where the factors are individually equated to zero in order to
find the roots r i, r 2, r 3, ...,rn.

306 Chapter 9 THEORY OF EQUATIONS
The degree of a polynomial equation is defined as that of
the highest-degree term of the polynomial. A first-degree
equation is called a linear equation, since its representation
in a coordinate system is a straight line. A second-degree
equation is a quadratic equation, a third-degree equation a
cubic equation, a fourth-degree equation a quartic equation, a
fifth-degree equation a quintic equation, etc.
A second-degree equation
x2 + px + q = 0
with the roots r\ and r2 can be written
(x-ri)(x-r2) = 0
or, developed,
x2 - 0*1 +r2) x + ri r2 = 0,
which gives
^1 + ^2 = ~P
rir2 = q
Similarly, a third-degree equation
jc3 +p x2 + qx + r = 0
with the roots r\,r2, and r3 is
fe-r1)fe-r2)fe-r3) = 0 ,
or
x3 - Oi + r2 + r3) x2 + Oi r2 + r2 r3 + r3 7*1) x - 7*1 r2 r3 = 0.
from which we have
^1 + ^2 + ^3 =-P ^
rir2 + r2rs+rsri = g I
The same kind of relationship exists between the roots and
coefficients of all polynomial equations of higher degrees in
monic form, with the coefficient 1 for the highest-degree term.
0 The coefficient of the term with the second-highest degree,
ra - 1, is the negative of the sum of the roots:
Tn = n + r2 + rz + ...+rn
0 The coefficient of the third-highest term, of degree n - 2, is
the sum of all double products of the roots:
Srjrj = r1r2 + rir3+... +rxrn _i + nrn + r2r3 + ... +r„_ir„
0 The coefficient of the fourth-highest term, n - 3, is the
negative of the sum of all triple products of the roots:
5>; rj rk = rir2r3+ ... +rn_2rn_irn
0 Etc.
0 The coefficient of the constant term of the polynomial is the
product of all roots,
Yin = nr2r3...rn,
with a plus sign if the degree of the polynomial is even and
a minus sign if the degree is odd.

307
Linear Equations
The simplest forms of equations are linear equations, in
which the unknown variable appears only in first-degree
terms.
A linear equation in two variables generally has no unique
solution unless there is some other condition that the variables
must satisfy.
Polynomial Equations
We distinguish two types of polynomial equations: linear
equations with integer coefficients and fractional equations.
Linear Equations with Rational Coefficients
The equation
3jc-(2jc + 1)-5 = 4jc + (4-2jc)-3jc
is solved in the following manner.
- Remove the parentheses:
3jc-2jc-1-5 = 4x + 4-2jc-3jc
- Transpose terms:
3jc-2jc-4jc + 2jc + 3jc = 4 + 5 + 1
- Combine:
2 x = 10; x = 5
Fractional Equations
Sy + 3 rt „., 53/ + 4
- Multiply both members of the equation by the least common
denominator, 5 • 8 = 40:
803/ + 243/ + 24-120 = 440-253/-20
- Transpose and combine terms:
80 3/ + 24 3/ + 25 3/ = 440-20-24 + 120
129 3/ = 516; 3/ = 4

308
Chapter 9 THEORY OF EQUATIONS
9.2 Equations with Absolute Values
Although first-degree equations in one variable generally
have only one solution, those containing an absolute-value
expression have more than one.
Examples:
|jc + 4| = 3
jc + 4 = 3 ; #1 = -1
-(x + 4) = 3; x2 = -7
\x + 2\ = |2jc-3|
jc + 2 = 2jc-3; jq = 5
jc + 2 = 3-2jc; #2 = o"
iHere once was a saCesman named 'Ness
'With high expenses, but sates much [ess.
He was known for his slumbers,
*But used absolute numbers,
So then he seemed one of the best.

309
Quadratic Equations
A polynomial equation of the second degree is known as a
quadratic equation,
ax2 + bx + c = 0,
where a x2 is the quadratic term, b x the linear term, and c the
constant term.
We distinguish between pure quadratic equations, without
the linear term, and complete quadratic equations, with
quadratic, linear, and constant terms.
A pure quadratic equation,
ox — oA = X ,
is easily solved; it reduces immediately to
2x2 = 32; x = ±4.
We shall describe three methods of solving a complete
quadratic equation: by factoring, by "completing the square",
and by employing the "quadratic formula".
In all three cases, we consider the quadratic equation reduced
to its monic form with the x2 coefficient = 1,
x2 +px + q = 0 ,
where p and q are real numbers.
Factoring
The equation
x2-x-2= 0
may be conveniently solved by factoring into
x2-x-2 = (x + l)(x-2) = 0,
where the binomial expressions are individually equated to
zero:
jc + 1 = 0 ; x =-1,
x-2 = 0; x = 2
Lost Roots
In factoring polynomials, there is a risk that a root may be lost
if both members of the* equation are divided by a factor that
contains the variable.

Chapter 9 THEORY OF EQUATIONS
The equation
x2 + x-2 = 4 x - 4
may be handled in two ways:
Right Wrong
x2-3x + 2 = 0
(x - 2) (x - 1) = 0
xi = 2; X2 = 1
(x - 1) (x + 2) = 4 (x - 1)
x + 2 = 4
x = 2
The method shown on the right loses us the root x = 1, since
division by (x - 1) is equivalent to dividing by zero, which is
not permissible.
A check of the solution does not discover a lost root.
Completing the Square
The idea of this method of solution is to make the binomial
expression
x2 +px
fp\2
a perfect square by adding a quadratic supplement ~ to
complete the square:
2 (P\2 ( PY
xz + px + I-J = Ix + 2 I •
To solve the equation
x2 - Sx = 4,
add — to both members of the equation,
2 q 9 a 9 ^
Xz - Sx +T=4+T = ~T~;
4 4 4
X 2) " \2) ; X " 2 ± 2 ;
xi = 4 ; X2 = - 1.
General Solution of Quadratic Equations
To solve the equation
ax2 + bx + c = 0,
bring it into monic form
x2 +px + q = 0,
or
x2 + p x = - q .
Complete the square by adding (p/2)2 to both members of the
equation, as before,
X+ 2) = T"9-

Section 9.3 Quadratic Equations
311
Quadratic Formula
Discriminant
Biquadratic Equation
By root extraction and rearranging the terms, we obtain as
solution the quadratic formula,
p , Vp2-4a
Y — — ""■■" "T" ^^~"""""~
2 " 2
or, for the general equation before reduction,
b Vfr2 - 4 a c
X = "2a ± 2a
The radicand in these expressions,
D= p2-4q or D= b2-4ac,
known as the discriminant, decides the nature of the roots of
the equation.
If
D > 0 the equation has two real, distinct roots;
D = 0 the equation has two real, duplicate roots;
D < 0 the equation has two conjugate complex roots.
To solve the fractional quadratic equation
x - 1 x - 2
1 -
x + 2 2jc-3 '
we multiply all terms by the least common denominator
(x + 2) (2 jc - 3),
Gc + 2) (2 jc - 3) - (x - 1) (2 jc - 3) = Gc - 2) Gc + 2),
which simplifies to
and
3(2jc-3) = x2-4
jc2-6jc + 5 = 0
x = 3±V9-5 = 3±2.
™ , 4 3
Check: #1 = 5; 1----=0
x2 = 1;
Both values are roots of the equation.
l-§-fl. 1-1.0
The name biquadratic equation is sometimes used as a
misnomer for the general quartic equation, but should correctly be
reserved for the special case
x*+px2 + q = 0,
which is actually a quadratic equation in x2, and is solved as
such.
xA-25x2 + l4A = 0
25 1 i f 16
x2 = — ± — V625 - 576 = <
2 2 | 9
*i,2 = ±4; *3,4 = ±3

312
Chapter 9 THEORY OF EQUATIONS
9.4 Inequalities
Conditional Inequality
Unconditional (Absolute)
Inequality
A conditional inequality is an inequality that is true only for
certain values of the variables. Thus,
(x + 3) > 5
is a conditional inequality, true only if x > 2.
An unconditional inequality, or absolute inequality,
(x2 + 1) > 0,
is true for all values of the variable.
Inequalities are solved in much the same way as equations.
5x-6>2x + 3; 3x>9; x > 3
The inequality is satisfied by all real numbers > 3 .
3 x - 6 5x + 1
; 9x-18>10x + 2;-x>20;x<-20
2 3
Changing the order of inequality members reverses the sign.
x2-3x-10 > 0
Inequalities with nonlinear members are solved by factoring:
(x + 2) (x-5) > 0
The numerical values of the factors must have the same sign;
that is, they must be both positive or both negative.
Two cases must be examined.
x + 2 > 0
x-5 > 0
x + 2 < 0
x-5 < 0
x > -2
x > 5
x <
x <
2
5
xi > 5
x2 < -2
x 2 - 3 x - 10 < 0 ; (x + 2) (x - 5) < 0
In this case, the numerical values of the two factors must have
opposite signs.
-2 < x < 5.
Impossible and
rejected.
x + 2 > 0
x-5 < 0 ,
x + 2 < 0
x-5 > 0
x >
>
X <
X <
X >
-2
5.
-2
5.

313
9.5 Root, Exponential, and Logarithmic Equations
Root Equations
Many equations require, for their solution, the squaring of
members of the equation, or parts of members; this procedure
may introduce extraneous roots, also called foreign roots,
belonging to an equation closely resembling the given
equation.
It is therefore often necessary, for these equations, to check if
the solutions obtained are, in fact, true roots of the given
equation; if not, they shall be rejected.
x-9 + V*-3 = 0.
Rearrange the terms and square both sides:
9-x = V*-3 ; 81-18jc+ x2 = x-S; x 2 -19 x + 84 = 0
x = \ (l9 ± V361-336) =|(19±5)= ^
Check: jc = 12; 12-9 + 3 = 6 A false root; rejected.
x = 7; 7-9 + 2 = 0 jc = 7 is a true root of the
equation.
V 5 x2 + 10 x - 6 = 2x + 3 .
Square both sides,
5jc2 + 10jc-6 = 4jc2+12jc + 9; jc2-2jc-15 = 0,
which factors to
(x - 5) (x + 3) = 0 ; x\ = 5 ; X2 = - 3 .
Check: x = 5; V125 + 50-6 = Vl69 = 13 , e. ^
y x = 5 is a true root.
25 + 3 = 13
x = -3; V45-30-6 = V9 = 3 1 * =-3 (toes not
S. satisfy the equa-
2(-3)+ 3 =-3 J tion; rejected.

314 Chapter 9 THEORY OF EQUATIONS
Exponential Equations
A number of exponential equations can be solved by algebraic
methods when the variable is part of an exponent; we can
distinguish three types of exponential equations:
o ax = an Since the bases are equal, the exponents must be
equal, and x = n (if a * 1).
o ax = 6* When different bases, both * 0, raised to the same
power are equal, the exponent must be = 0.
o ax = bn Both bases are assumed to be * 0, and n * 0; the
equation is solved by taking the logarithm of both
sides, to the same base:
Igb
x • lg a = n Igb ; x = n
\ga
a 6 = a z ; o
325* = 2003 ; x • lg325 = 3 • lg200 ; x = 3 •. gf^ .
° ° lg325
3.32^.-.24^.25.5^ = Q.22xS2xf
which simplifies to
24
22x & = %•
Take the logarithm of both sides,
lg 0.32
x = f _„ = -0.380 352 5....
Ig20
S2+x + S2-x = 82.
9.3* + _= 82; 9-3^-82-3^ + 9 = 0,
which can be factored to
(9-3*-l)(3*-9) = 0
3* = |; x = -2
Sx = 9; x = 2

Section 9.5 Root, Exponential, and Logarithmic Equations
315
Logarithmic Equations
Since logarithms are a form of exponent, there is no clear
distinction between exponential and logarithmic equations, or
between the methods of solving them.
*lg*-l = 100.
Take the logarithm of both sides,
(Igx - 1) Igx = 2
lg^x-lgx-2 = 0 ; (Igx+ 1) (Igx-2)
= 0
\g x = - 1
x\ = 0.1
Igx = 2
x2 = 100
lg (2 x2 - 1) - 2 • lg (2 + x) = 0 .
Rearrange terms and equate antilogarithms,
2 x2 - 1 = (2 + x)2
rewritten,
2 x2 -1 = 4 + 4 x
x2 — 4 x - 5 = 0,
+ jc2, or
(jc+1)(jc-5) = 0; #1==-1; #2 = 5.
Check: xi = -1; lgl-2-lgl = 0-0 = 0,
x2= 5; Ig49-2 1g7 = 21g7- 2 • lg 7
Both roots belong to the original equation.
In (3 x - 8) + 2 = V2 • In (3 x - 8) + 7 .
Substitute In (3 jc - 8) = u ,
= 0
u + 2 = V2z7+7 .
Square both members,
u2+ 4:11 + 4 = 2u + l
u2 + 2u-3 = 0
(w - 1) (u + 3) = 0
In (3*-8) = 1
3jc-8 = e
xi = 3 (e + 8) * 3.573
ln(3x-8) = -3
3jc-8 = e~3
(x2 = 3 (e-3 + 8) * 2.683J
By squaring the members of the equation, one runs the risk of
acquiring an extraneous root. Checking the roots one finds
that only x\ satisfies the original equation; x2 is an extraneous
root.

316
Chapter 9 THEORY OF EQUATIONS
9.6 Cubic Equations
Historical Notes
HERON
(1st century)
KHAYYAM Omar
(10487-1122)
ARCHIMEDES
(c. 287 - 212 B.C.)
del FERRO Scipione
(1465 -1526)
TARTAGLIA Niccolo
Italian, tartaglia, "stammerer'
(c. 1500 -1557)
CARDANO Girolamo
(1501 -1576)
The earliest occurrence of cubic (third-degree) equations may
have been in antiquity in connection with the famous problems
of duplicating the cube and trisecting the arbitrary angle,
which both would lead to cubic equations if stated analytically.
The task was to solve these problems using only an unmarked
ruler and a pair of compasses, which we now know is
impossible. However, several mathematicians of antiquity
solved them geometrically using other curves as tools.
An important advance was made by the mathematician-
inventor Heron of Alexandria by reviving and developing old
Babylonian and Egyptian practices of extracting roots by
successive approximation.
The cubic equation was a cherished topic of study among
mathematicians of the Muslim world. Best known is the work
of the Persian poet, mathematician, and astronomer Omar
Khayyam. Building on the Greek tradition, he obtained
solutions to cubic equations as intersections between conic
sections; such studies had, however, been made by Archimedes
about 1000 years before Khayyam. In mathematics, the
importance of Omar Khayyam and other early Muslim
mathematicians lies mainly in their perpetuation of ancient Greek and
Hindu knowledge.
Outstanding among mathematical discoveries during the
Renaissance were the general solutions of the cubic and
quartic equations by radical expressions - an emancipation
from a 2000-year-old framework of knowledge set up by the
Greeks, Indians, Arabs, and Persians.
About 1515 Scipione del Ferro, a teacher of mathematics at the
University of Bologna, solved a cubic equation lacking the
quadratic term,
jc3 + q x = r .
After the manner of his time, he kept his discovery secret,
except for telling it in confidence to one of his pupils, Antonio
Fiore.
Some time around 1535 Niccolo Tartaglia, a mathematics
teacher in Venice, solved a cubic equation in another form,
without the linear term,
jc3 + q x2 = r .
He, too, guarded his solution as a dead secret until he divulged
it, in the form of a cipher in verse and against a solemn
promise of silence, to Girolamo Cardano, of Milan and Pa via,
a mathematician of note and a famous physician.
In 1545 Cardano published his great work
Artis magnae sive de regulis algebraicis
- generally known as Ars magna - in which, breaking
his promise, he included Tartaglia's solution of the cubic
equation.

Section 9.6 Cubic Equations 317
Solutions of Cubic Equations
The general form of a cubic equation,
ax? + bx2 + cx + d = 0,
can be reduced to the normal form,
s?+px* + qx + r = 0,
where jc3 is the cubic, p x2 the quadratic, q x the linear, and r the
constant term.
Every cubic equation with real coefficients has at least one
real root r\; the other two roots may also be real or a conjugate
complex pair. The cubic expression can be written as a product
of a binomial (jc - r{) and a second-degree polynomial which
gives the roots r2 and r% when equated to zero.
A pure cubic equation, jc3 +1 = 0, has the three roots
3T—
T2 = W2^-t
where
3r-
w2 = | (- 1 + i <S)
w3 = | (- 1 - i V3)
(c/ jop. 793 - 94) are the complex cubic roots of unity.
A symmetric cubic equation,
ax? + bx2 + bx + a = 0,
can be reduced to monic form,
jc3+pjc2+pjc+1 = 0,
and factored to
(jc + 1) (jc2 - x + 1 + p x) = 0 .
Consequently, the equation has a root r\ = -1. The other two
roots are obtained by equating the polynomial (jc2 - x + 1 + p x)
to zero; to solve that equation, we write
x2 + (p-l)x +1= 0.
Then, using the quadratic formula, we find
p-1 , Vp2-2p-3
r2,r3 = ——± 2 *
Example: y3 + 6y2 + 6y + l =0
(y + l)(y2 + 5y + l) = 0
y\ = -1
^2,^3 = ~2 ~2~*

318
Chapter 9 THEORY OF EQUATIONS
General Solution of Cubic Equations
The cubic equation in general form,
ax3 + bx2 + cx + d = 0,
can be transformed by substituting
x=y-
3a
into the reduced form,
y3+py + q= 0.
We assume that this equation has a root
y = u + vy
which, inserted in the cubic equation, gives
(u3 + v3) + (3uv +p)(u + v) + q
This equation is satisfied if
= 0
3uv = —p;
^-^ = (- |
p\3l
J
ifi + v3 = -q
Thus, u3 and v3 are the roots of a quadratic equation,
r3l
fl+at-lZ
\s
7
= 0;
w
V
-*WR
\2 /p\3
u and v being interchangeable, we select
u\
-aTiWI
-^)
>
^i
-^-l-<[i
J
'P\
+ l3
J
u2
"2
V1OO3 f v3
uicos
V1CD2 I,
y
where 0*2 and ^3 are the complex cube roots of unity,
<*>2> fl>3= 2 (~l±iV3) .
The three solutions of the original equation are then
b "\
xl = u>i + vi~
X2 = U2 + V3-
3a
3a
xs = us + v2- —
j

Section 9.6 Cubic Equations
319
The radicand
■> ■ 'IN?'
Discriminant
VIETE Francois
(1540 -1603)
is the discriminant that determines the nature of the roots of
the reduced equation.
If D > 0 we have one real root, and
two conjugate complex roots;
D = 0 we have three real roots, of which at least two are
equal;
D < 0 we have three distinct real roots.
To solve a cubic equation whose discriminant is negative, one
must extract cube roots of complex numbers. For a long time
all attempts to solve, algebraically, such an equation - the
casus irreducibilis — came to naught despite the fact that this is
the very case when the equation has only real roots. This
deadlock was resolved by trigonometric means around 1600
by Francois Vi&te.
A particularly elegant solution of the cubic equation was
presented by Viete in De aequationum recognitione et emenda-
tione, published posthumously in 1615.
In the cubic equation without the square term
x? + 3ax = 26
he substituted
a
x = — - y ;
y
the equation then becomes
y6 + 2by3-as = 0,
that is, a quadratic equation in y3; solve for y3, extract the cube
rooty, and compute the x values.

320
Chapter 9 THEORY OF EQUATIONS
9.7 Quartic Equations
FERRARI Ludovico (or Luigi)
(1522 -1565)
al-KASHI Masud
(? -1429)
The general solution - with rational operations and radical
expressions - of the quartic equation proposed by Ludovico
Ferrari is, as we shall see, rather complicated and
cumbersome because it relies, in several steps, on the solution of
equations of lower degrees.
By a stroke of good luck, Nature has ordained that problems
presenting themselves in physics seldom lead to general
quartic equations; the quartic equations that do occur - for
instance, the irradiation of light and heat from a black body -
are of a kind more easily coped with, numerically or
graphically, if need be.
Actually, the enthusiasm raised by the 16th-century solutions
of the cubic and quartic equations was of no practical
significance, since the Persian al-Kashi - a century before
del Ferro, Tartaglia, Cardano, and Ferrari - could have
solved any cubic equation to any desired degree of accuracy by
successive approximation.
The significance of the del Ferro/Tartaglia/Cardano/Ferrari
formulas was primarily their contributions to an accelerated
development of algebra.
The Symmetric Quartic Equation
The equation
2^-9^ + 14^-9^ + 2 =0
is solved by grouping terms with like coefficients together and
dividing all terms by the square of the variable,
2 x2
x2 )
-9U + -
x )
+ 14 = 0,
which we rewrite
x2 + 2
x2 )
\
9 1
--he + —
2 V XJ
or
1
x + —
X J
\2
9
\
2 V xj
+ 5 = 0,
+ 5 = 0,
n
that is, a quadratic equation in the new variable he + — , with
x
J
the solutions \x + —
V XJ
1 5
x + — = 77
x 2
4 ~ \ 16
5 :
x2-—x + l = 0
x\ — 2, X2 — o
* + -= 2
x
x2-2x+\ = 0
*3,*4 = 1

Section 9.7 Quartic Equations 321
The near-symmetric equation
jc4-8jc3 + 14jc2 + 8jc-8 =0
can be rearranged as
(jc4-8jc3 + 16jc2)-2(jc2-4jc)-8 = 0,
that is, o o o
(x2-4x)2-2(x2-4x)-8 = 0,
a quadratic equation in the binomial y = (x2 - 4 x) with the
solution A
r- 4
'2-2y-8 = 0; y = 1±V9 = 1±3 =
y-zy-s = u; y = i±yv = i±j =_2
x2-4x + 2 = 0
jc3,jc4= 2±^2
jc2-4jc-4 = 0
xl9x2 = 2±2^2
Solution by Factoring
Certain quartic equations may be reduced to equations of a
lower degree, generally quadratic equations, by factoring.
If one or more roots of the equation are known, or easily
identified, we may reduce the quartic equation by dividing out
the known root.
jc4-6jc3 + 5jc2 + 12jc-12 = 0.
We note that the equation is satisfied by x\ = 1 and x\ = 2,
and divide the fourth-degree polynomial by the product
(x - 1) (x - 2) = (x2 - 3 x + 2), which gives the equation
x2 - 3 x - 6 = 0;
that is, we have reduced the original polynomial to a product of
two quadratic polynomials,
(x2 - 3 x + 2) (x2 - 3 x - 6),
which we equate to zero separately:
jc2-3jc + 2 = 0
xi = 1, X2 = 2
jc2-3jc-6 = 0
*3,*4 = o (3±V33)
Solution by Rearranging
• jc4 + 3jc3 + 12jc-16 = 0.
The equation can be rearranged as
(jc4-16) + (3 jc3 + 12 x) = 0
or (x2 + 4) (x2 - 4) + 3jc(jc2 + 4) = 0
and
(x2 + 3 x - 4) (x2 + 4) = 0.
jc2 + 3jc-4 = 0
x\ = 1, x2 = -4
x2 + 4 = 0
JC3,JC4 = ±2i

322 Chapter 9 THEORY OF EQUATIONS
jc4-9jc2 + 12jc-4 = 0.
The equation can be rearranged as
x4 - (9 x2 - 12 x + 4)
= x4- (3x-2)2 = (x2 + 3x-2)(x2-3x + 2) = 0
jc2-3jc + 2 = 0
X\ = 1, #2 = 2
jc2 + 3jc-2 = 0
Vl7
XS ,x4 ~~ Q
General Solution of Quartic Equations
The Kindling Proposal
In 1540 a mathematics dilettante, Zuanne de Tonino da Coi -
also known as Ioannes Colla - proposed this problem,
"Divide 10 into three parts such that they shall be in
continued proportion and that the product of the first two
shall be 6",
which he gave to Cardano, who, himself unable to solve it,
handed it to his pupil Ludovico Ferrari, who could - and did.
The problem can be stated,
a + b + c = 10; alb = b/c; ab = 6,
which gives
a = 6/6; c = b2/a = 63/6,
and thus,
6 63
- + b + — = 10
b 6
64 + 662-606 + 36 = 0,
a quartic equation, without a third-degree term, which Ferrari
rewrote,
(62 + 6)2 = 662 + 606,
and, again,
(62 + 6+z)2 = (2z + 6)-62 + 60 6 + (12z+z2).
For the right-hand member of the equation to be a perfect
square, the discriminant of the quadratic polynomial must
equal zero,
4-(2z + 6)-(z2 + 12z) = 602;
this gives the resolvent cubic equation
z3 + 15z2 + 36z = 450,
which appears in Cardano's Ars magna thus:
habebimus 1 cubum p: 15 quadratis
p: 36 positionibus aequada 450.

Section 9.7 Quartic Equations 323
By substituting
z = y-5
we have
y3-S9y-S80 = 0.
A comparison with
p. 318 y^+py + 9 = 0
gives:
p = -39 p/3 =-13 (p/3)3 = -2197
9 = - 380 9/2 = - 190 (9/2)2 = 36 100
33 903
3 3
y = V190+ ^33^03 + Vl90 - ^33903 * 9.0098 * 9;
z * 9-5 = 4.
Inserting 2: ** 4 in
(62+ 6+z)2 = (2z +6)-62 + 606 + (122+22)
we obtain
(62+10)2 * l462 + 606 + 64,
whose right-hand member must be a perfect square,
m2 62 + 2 mn • 6 + n2 = (m 6 + n)2
m ** Vl4 ; n ** V64 = 8.
We now have
62+10 * 6^14 + 8; 62-6 Vl4 + 2 * 0
6 ^-^-(+) JL"2 = 2 V^ + V6j * 3.09557;
12
a * 6/6 = * 1.938 25;
Vl4 + V6
(Vl4 + V6)
alb = 6/c => c = 63/a,
3
c = — * 4.943 93 .
48
Checking:
a + 6 + c = 1.938 25 + 3.095 57 + 4.943 93= 9.98 * 10
„ 1.938 25 _rt„ t. 3.095 57 „„^
a/6 = ^0953^ 06261; 6/c =194^98- " °6261
a • 6 = 1.938 25 • 3.095 57 = 5.99988... * 6

324
Chapter 9 THEORY OF EQUATIONS
pp. 322 - 23
General Formula for Solution of Quartic Equations
The general quartic equation
ax* + bx^ + cx2 + dx + e = 0
can be reduced by substituting x =y -b/4a and dividing by a to
the form
y*+py2 + qy + r = 0,
a quartic without a third-degree term, which is solved as
above.
Rewriting
(y2+p)2 _ py2 -qy + (p* - r)
and
(y2+p+z)2 = (p + 2z)y2-qy + (p2-r + 2pz + z2),
where the discriminant of the quadratic expression is equated
to zero,
4 • (p + 2z) • (p2 - r + 2p z + z2) = q2,
gives the resolvent cubic equation
,3_H
z6 --pzl + (2pl-r)z + -\p6 -pr - —
r2\
= 0.
y
We may check this equation using Colla's proposal, and have
64 + 662-606 + 36 = 0; thus,p = 6; q = -60; r = 36.
23.1^2 + (2.62-36)2 + 1(^3-6. 36 _L-|2)2^
= 0,
J
or zs + 15z2 + 36z = 450,
which agrees with the problem on p. 322.
If the roots of the resolvent
cubic equation are
all real and > 0
all real, one > 0, two < 0;
all real, two > 0, one < 0;
or all real, three < 0
one real, two conjugate
complex roots
the roots of the quartic
equation will be
all real;
two pairs of conjugate
complex roots;
two real roots, one pair
of conjugate complex roots.
Jor Cardano, rnatfiernatician-pfiysician,
'Equations were the greater ambition.
'But zuith his crystal ball hazy,
And his patron quite crazy,
He gave typhoid its modern description.

325
Systems of Equations
Solving an equation in more than one variable usually calls
for additional conditions being imposed on the variables. For
a first-degree equation, e.g.,
3x-4y = 3,
the number of pairs of x and y that satisfy the equation is
unlimited.
For a unique solution to exist, yet another equation with the
same variables is required, e.g.,
x + 2y = 11,
which gives us the common solution
x = 5 ; y = 3 .
In a system of simultaneous equations, all of the equations
must be true at the same time. A complete solution with
identification of all roots of the several equations is possible
only if the system has as many equations as there are
variables.
We distinguish three cases:
o The system has a unique solution, as above.
o The system is not uniquely solvable, but has infinitely
many solutions:
6x + 8y = 201 3jc + 4y = 10
3(*-l) + 4y = 7 J ^ 3x + 4y=10
o The equations of the system may be contradictory; the
system then has no solution:
6x + 8y =241 3x + 4y=12
3(*-l) + 4y = 7 J ^ 3jc + 4y = 10
Systems of Linear Equations
To solve a system of two linear equations in two variables,
3x + 2y = 19
2x-3y = 4
we have a choice of three methods: adding, substituting, and
equating.

326 Chapter 9 THEORY OF EQUATIONS
Addition
Multiply each equation by a suitable quantity so as to make the
coefficients of the two variables agree between them; adding
or subtracting the resultant two equations eliminates one
variable.
•3 => 9x +6y = 57 or -2 =
•2=>4x-6;y=8 or • (-3) =
13 x
Substitution
= 65
x = 5
y = 2j
6x
- 6 x
+ 4y
+ 9y
13y
—
= .
=
X
y
38
-12
26
= 5
= 2j
Solve one equation for one of the variables and insert this
value in the other equation.
2*-4 n 2 (2*-4) _
y=—3—; 3x+ - =19;
9jc + 4jc-8 = 57; 13 jc = 65; etc.
Equating
Solve both equations for one of the variables and equate the
solutions:
19-3* 2jc-4
y = = —^ ; 57 - 9 x = 4 x - 8 ; 13 x = 65 ; etc.
Practical problems often lead to systems of linear equations in
many unknowns. To solve such problems a method called
Gaussian elimination is convenient; it will be described in
p. 661 Chapter 18, where we employ matrices and determinants to
facilitate the solution of unwieldy systems of linear equations.

Section 9.8 Systems of Equations
327
Systems of Linear and Quadratic Equations
From the multitude of systems of equations that include
nonlinear equations, we select two frequently occurring types.
One Quadratic, One Linear Equation
(1) :c2+y2 = 26
(2) x -y = 4
y = jc-4
Inserting (2) in (1),
x 2 + (x - 4)2
2jc2-8jc + 16
x2 — 4x — 5
(x -5)(x + 1)
x\ = 5
*2
^2
= 26
= 26
= 0
= 0.
= -1
= -5
Two Linear Equations, One Quadratic
(1) x-2y-z = 14
(2) 2x-y-z = 18
(3) x2+y2+z2 = 84 J
Using the linear equations, express x andy in terms of z:
x =
(1) x-2y-z =14
(2) 4x-2y-2z =36 J
(1) 2:c-4y-2z = 28
(2) 2x-y-z = 18
Insert x and y in (3):
z + 22\2 ( z + 10\2 9 _
+ I - —-— I + z2 = 84
z + 22
3
z + 10
y = -
z2 + 44z + 484 + z2 + 20z + 100 + 9z2 = 756
2 W 172 n
32
z = -— +
11 "
V2916
11
32 ±54
11
zx = 2
jci = 8
yi = -4
^v
>
J
*2 = -
X2 =
^2 =-
86
11
11
_8_
11
>
J

328
Chapter 9 THEORY OF EQUATIONS
Systems of Two Second-Degree Equations
(1) x2+y2+x+y = 8
(2) xy + x + y = 5
The following approach is one of several possible.
Multiply (2) by 2 and add to (1):
(x+y)2+ 3 (x+y)-18 = 0
(x+y) = \ (-3±V9 + 72) = ±(-3±9) = «
f 3
-6
x+y = 3
xy = 2
which lead to
x +y = -6
xy = 11
jc2-3jc + 2 = 0 | jc2 + 6jc + 11 = 0
and the solutions
x1 = 2
*2 = i
y2 = 2
*3>*4 = -3±iV2
^3,^4 = -3 +i V2
Systems with One Homogeneous Equation
All terms in a homogeneous equation have the same degree,
which makes it possible to determine the ratio of the variables.
Consider a system of equations in two variables; one of the
equations is homogeneous:
(1) 6x2 — xy—y2 = 0
(2) x2-2y =27
Substitute
in (1),
y = ex
x2 (6 - c — c2)
c2 + c -6 :
ci = 2;
Insert in (2),
y = 2x
x2-±x-21 = 0
= 0
= 0;
C2
= -3
y = -3*
jc2 + 6jc-27 = 0
*l,x2 = 2±J31
yi,y2 = 4±2aT31
X3 = 3
^3 = - 9 .
X4 = -9
; y* = 27

Section 9.8 Systems of Equations
329
Systems with Symmetric Equations
Variables in symmetric equations are interchangeable. The
simultaneous equations
(1) x2+y2 = 13
(2) xy = 6
are symmetric and homogeneous.
As with most systems of equations, the solution can be arrived
at by different approaches; below, we account for two.
Insert y = — from (2)
J x
into (1):
A
x2 +' —
= 13
:^-13:^ + 36 = 0
(x2-9)(x2-4) = 0
x = ± 3
6
y =±o
x = ± 2
6
y =±o
Multiply (2) by 2 and add to (1):
x2+y2 + 2xy = 25
{x+y)2 = 25
x +y = 5
xy = 6
x (5 - x) = 6
jc2-5jc + 6 = 0
0c - 3) Gc - 2) = 0
jc +y = -5
xy = 6
x (- 5 - jc) = 6
jc2 + 5jc + 6 = 0
(x + 3) (jc + 2) = 0
^,^2 = ±3
yi,y2 = ±2
*3>*4 = ±2
^3,^4 = ±3

330
Chapter 9 THEORY OF EQUATIONS
9.9 Diophantine Equations
DIOPHANTOS
(c. A.D. 250)
Euclidean Algorithm,
GCD: p. 126
Diophantus of Alexandria carried out extensive studies of
problems relating to indeterminate equations, which he
published in Arithmetica; this work may have consisted,
originally, of thirteen books, six of which survive through
Arabic translations.
Diophantus accepted any solution in rational numbers, but the
name Diophantine equations today refers exclusively to
equations with integer solutions.
The prototype equation
ax+by+c = 0
has integer solutions only if c is a multiple of the GCD (a, 6). It
is solved for one of the variables, preferably the one that has the
lowest coefficient or whose coefficient is a factor of the constant
term; one can then determine integer values of the other
variable.
Find the positive integer values of x and y that satisfy the
equation
12x+ ly = 220.
The GCD of 12 and 7 is 1. We can solve the equation either for
x or for y:
220-7y 7y-4
x = — = 18 -
12
12
7y-4
where ——— must be an
integer < 17, that is,
0<y<29 with Ay = 12.
The lowest possible y is 4,
as 4 • 7 - 4 = 24 = 2 • 12.
220 - 12 x M 12 x - 3
y= -7 -si-—5—
12 x — 3
where = must be an
integer < 30, that is,
0<jc<18 withAx = 7.
The lowest possible x is 2,
as212-3 = 21 = 3 7.
jci = 2
y\ = 28,
x2 = 9
J2 = 16,
#3 = 16
ys = 4 ,
Find all natural numbers x and y that satisfy
2x + 5y = 32.
Solving for x gives
x = 16 -
5y
32 2
x is > 0, if y < — = 6 —, and even.
y\ = 6
*i = i.
y2 = 4
; X2 = 6 .
1 ys = 2 ]
; *3 = 11.

Section 9.9 Diophantine Equations
331
SVRCtiA'EOLOgiCSlC
sociiny
h
Established 1873
The membership of an archaeological society is limited to 100;
members pay an annual fee of $125, reduced to $75 for senior
members of 60 years of age and older. One year, the total fees
paid by members below 60 exceeded those paid by senior
members by $10 575.
How many members were under 60 years of age, and how
many were older?
If members under 60 number x, senior members y, we have
125x -75y = 10 575
or, simplified,
from which
Fory > 0, we must have
and x divisible by 3:
x1 = 87 '
x2 = 90
,
m
>
5x-Sy = 423,
5 x - 423
^=3 "
,
423 3
x > -=- = 84 -
5 5
y\ = 4 '
^2 = 9 J
87 + 4 = 91 < 100
> 90 + 9 = 99 < 100
The society had either
91 members, 87 of whom were below 60 years of age and 4
aged 60 or more, or
99 members, 90 below 60 and 9 senior members of 60 or
more.
In the equation
xy - 5jc + 4y = 0,
the variables x and y are natural numbers or zero; solving for
y gives
y =
5 x
x + 4
= 5-
20
x + 4
y is a positive integer, or zero, only if
20 c
7 * 5>
x + 4
where x is a positive integer or zero. We find the following
solutions,
Xq = 0
yo = o
X\ = 1
; y\ = i
*2 = 6
1 ; y* = 3 J
#3 = 16
; ^3 = 4
of which the solution xq yo is trivial.

332 Chapter 9 THEORY OF EQUATIONS
Eight students were given a list of synonymous words that
were to be sorted into pairs. Within the allotted time, the best
student had 21 correct pairs; the next best three students had
two-thirds of the possible pairs each; and the remaining four
students each had half the possible number of pairs plus four.
What was the total number of possible pairs?
With N as the sought number, we have
21 2
N > 3
\n>\n+± J
63
► giving 24 < N < — .
N is an integer divisible by 2 and 3; thus, the total number of
possible pairs of synonyms was 30.
The Pythagorean Number Theorem
Pythagorean Numbers Pythagorean numbers, or triples, consist of positive integers
Pythagorean Triples a, b, c, satisfying the relation
a2 + b2 = c2;
e.g., 3, 4, 5 form the Pythagorean triple of 32 + 42 = 52 .
Basic Pythagorean triples have no common factor; thus, every
Pythagorean triple is basic or a multiple of a basic triple.
Analysis of Babylonian tablets dating before 1600 B.C. by Otto
Neugebauer and his co-workers shows that not only did the
scribes of that era know the relation a2 + b2 = c2 but they also
knew how to find all triples of integers satisfying it.
To obtain Pythagorean triples one may choose any odd positive
number and divide its square into two integers that are as
equal as possible in size; e.g., 92 = 81 = 40 + 41, and the triples
are 9, 40, and 41. One may also choose any even positive
number h; the Pythagorean triples are then 2h, h2 - 1, and h2 + 1.
The Pythagorean number theorem gives all the basic triples
of positive integers a, 6, c, having no common factors, and
satisfying
a2 + b2 = c2,
such that
a = 2 mn ; b = m2 - n2 ; c = m2 + n2,
where one but not both of m and n must be even, and m and n
are positive integers with m > n > 0.
Why? If m and n both were even, then a,6, c would all be even, which is
ruled out by the premise of no common factors; and if both were
odd, a2 + b2 would have a remainder of 2 on division by 4, while
c2 must have remainder 0, which is impossible.
Suppose b is odd; move it to the right side, and factor to get
a2 = c2-62 = (c-6)(c + 6),
where (c - b) and (c + 6) must have in common only a single
factor of 2, otherwise c and b would have common factors.

Section 9.9 Diophantine Equations
333
But then (c - b) I 2 and (c + b) I 2 must be perfect squares; the
only possibility is that
c-b = 2n2; c + b = 2 m2
for some integers m and n without common factors and
satisfying m > n > 0. So all basic Pythagorean triples are given by
a = 2 mn ; b = ra2 - rc2 ; c = ra2 + rc2,
with m and rc as above. Substitution shows such triples satisfy
a2 + 62 =c2.
This result is given in Euclid's Elements (c. 300 B.C.).
Fermat's Last Theorem
de FERMAT Pierre Fermat's last theorem, also called Fermat's great theorem, has
(1601-1665) been one of the most famous mathematical conjectures.
In 1637, in the margin of a copy of Diophantus's Arithmetica,
Fermat stated - in our notation - that
if n > 2, the equation
xn + yn = zn
cannot be solved in positive integers x, y, z.
Fermat's last theorem is an impossibility theorem.
Fermat claimed, "and I have assuredly found a marvellous
proof of this, but the margin is too narrow to contain it".
We know that Fermat had a proof for the case n = 4, but he did
not mention a general method of proof when he wrote down the
proof for n - 4.
When the theorem once has been proved true for n = 4, we only
need to prove the case where n is an odd prime, because in
xn + yn = = zn, n is either (I) a power of 2 or (II) divisible by an
odd prime p.
Why? I:
If n = 4k, where k is a natural number, we have x*k + y*k
= z4k, which is the same as
(1) (xkf + {yk)A = (zk)A.
Substitute a = xk, b = yk, c = zk. If there existed a triple
of natural numbers (jc, y, z) satisfying (1), then a4 + 64 = c4
would also exist, which is impossible since the (impossibility)
theorem is true for n = 4.
II:
If n =pk, where p is an odd prime and k a natural number,
then x n + y n = zn may be written
(2) (xk)p + (yk)p = (zk)p.
Substitute a = xk, b = yk, c = zk. If there existed a triple of natural
numbers (x, y, z) satisfying (2), then a,P + b? = cP would also
exist.

Chapter 9 THEORY OF EQUATIONS
EULER Leonhard
(1707 -1783)
LEGENDRE Adrien Marie
(1752 -1833)
LAME Gabriel
(1795 -1870)
GERMAIN Sophie
(1776 -1831)
WILES Andrew
(b. 1953)
TAYLOR Richard Lawrence
(b. 1962)
Euler proved Fermat's last theorem for n = 4 and attempted,
around 1753, a proof for the validity of the theorem for
exponents of 3, but some details had been left out; later, Gauss
gave a correct proof. In 1825 Legendre proved the theorem for
exponents of 5 and, in 1839, the French mathematician and
engineer Gabriel Lame for exponents of 7.
With computer technology Fermat's theorem has been proved
for exponents less than 4 000 000 by Joe Buhler and Richard
Crandall; such investigations cannot tell us whether the next
exponent might falsify the theorem.
The French mathematician Sophie Germain found some
results on more general cases; she showed that if n is an odd
prime < 100, the equation xn + yn = zn is insolvable in integers
not divisible by n. In 1941, D. H. Lehmer and E. Lehmer
extended Germain's theorem to all primes less than 253 747 889.
Until recently all claims to a general proof of Fermat's last
theorem proved false, but in June 1993 the Englishman
Andrew Wiles of Princeton University gave the essential
steps toward such a proof. The proposition that Wiles set out to
prove was, however, not Fermat's but one of much greater
scope, to which the proof of Fermat's last theorem came only as
an incidental consequence. Wiles applied methods of algebra
and geometry; he made no use of computers. A gap in the
argument of the alleged proof was soon revealed, but, in
October 1994, Wiles, with the English mathematician Richard
Taylor, filled the gap and gave a proof of certain crucial
properties of the algebras used in Wiles's general proof. The
completed proof, more than 130 pages, was published in May
1995 in the Annals of Mathematics. Although some caution
was still recommended in 1996, most experts are confident that
the Wiles-Taylor proof will withstand further scrutiny.
If Wiles and Taylor have succeeded, a great advance in
number theory has been accomplished, but it will not end the search
for the other proof - the one that Fermat had "assuredly found"
about 360 years ago. Experts seem to believe, however, that a
relatively brief proof of Fermat's last theorem simply isn't
attainable.
- Only marvellous, not brief.
Mon Dieu! I told you long ago it
wouldn't fit in the margin!
Messieurs Wiles et Taylor,
mes compliments!

335
Chapter
10
INTRODUCTION TO FUNCTIONS
Page
10.0 Historical Notes 336
10.1 Groundwork 336
10.2 Elementary Functions 348
10.3 Continuity and Limits 355

336
Chapter 10 INTRODUCTION TO FUNCTIONS
10.0 Historical Notes
von LEIBNIZ Gottfried
(1646-1716)
EULER Leonhard
(1707-1783)
FOURIER Joseph
(1768 -1830)
DIRICHLET
Peter Gustav Lejeune
(1805 -1859)
The term "function" - Latin functio — first appeared in 1692 in
a mathematical article in the Acta Eruditorem to denote
various tasks that a straight line may accomplish with respect
to a curve, such as forming a chord, tangent, or normal. The
article was signed O.V.E. but is attributed to Gottfried von
Leibniz, the German mathematician. In an article from 1694,
also in Acta Eruditorem, Leibniz gave the term "function" a
more specific meaning by letting it denote the slope of a curve,
a definition that has very little in common with the present-
day mathematical definition of a function.
The Swiss mathematician Leonhard Euler in 1749 defined a
function as a variable quantity that is dependent upon another
quantity, thereby approaching today's definition.
Euler's definition was challenged when the French physicist
and mathematician Joseph Fourier in 1822 presented his work
on heat flow (Theorie analytique de la chaleur). For his
investigations, Fourier introduced series with sines and cosines as
terms, which led to the concept that a given representation of a
function may be valid for only a certain range of values.
Based on Fourier's investigations, the German
mathematician Lejeune Dirichlet in 1837 proposed that, from the
mathematical point of view, a function is a correspondence that
assigns a unique value of the dependent variable to every
permitted value of an independent variable. There will be
reason to return to this definition many times in this text.
10.1 Groundwork
Independent Variable,
Argument
Dependent Variable
Map, Mapping, Operator,
Transformation
When we buy coffee and pay by mass, we can say that the price
is a function of mass. If the cost of coffee is $9.00 per kilogram
(kg), half a kg is $4.50, 2 kg is $18.00, etc. We have the
formula
f(x) = 9x,
where x designates the purchased amount (kg) and fix) is the
price (in $). Here, fix) is defined only for x > 0.
Expressed in a more mathematical way, we say that a function
is an association between two or more variables, in which
to every value of each of the independent variables, or
arguments, corresponds exactly one value of the dependent
variable in a specified set called the domain of the function.
Map, mapping, operator, and transformation are other names
for a function.

Section 10.1 Groundwork
337
STORM PETERSEN Robert © storm p.- museum, Copenhagen
(alias Storm P.; 1882-1949)
A function of one variable x may be written
fix),
which reads "fofx" or, in full, "the value of the function f at x";
a function of several independent variables is written,
analogously,
T \X9 y, z ...) .
Letters of the Latin or Greek alphabet may be used to designate
functions; the letters f F, g, G, h, H, u, u, and (p (phi), y/ (psi)
are the most common.
It is standard practice to write the dependent variable on the
left-hand side of the equality sign of an equation; thus, in
y = x + l or fix) = x + 1,
y or f(x) is the dependent variable, x the independent variable.
On the other hand, in the expression
x=y + 1 or fiy)=y + 1,
x or fiy) is the dependent variable, y the independent.

338
Chapter 10 INTRODUCTION TO FUNCTIONS
Orthogonal Coordinate System:
p. 472
X
The equation x + y = 1 denotes a straight line, as shown by the
graph; the line may be represented by either of the two
functions
fix) = 1-x )
fiy) = l-y
*~ function of y
- not a function
The above curve represents the equation
x =y2.
There is only one value of x for a value assigned to y, which
means that x is a function ofy.
If, instead, we consider y to be dependent on jc, we solve the
equation for y, obtaining
y = ±V* ,
which shows that there are two values ofy for every value ofx,
indicating that y is not a function of x.
Restricting the equation to either
y = yx or y = -*yx ,
representing the upper and lower half of the parabola,
respectively, there is only one value of x for every y,
f+ (x) = V*; /1 (x) = - V* •

Section 10.1 Groundwork
339
Explicit and Implicit Forms
The equations - and functions -
y = fix) = 2jc + 5 and z = g(x) = Sx-x2f
where the dependent variable is given in terms of the
independent variable, are said to be in explicit form.
In contrast, an implicit form is characterized by the
occurrence of dependent as well as independent variables on one
side of the equality sign:
2x-y = 6; x2+y2 = 9.
The above equations may be rendered into explicit form,
y = 2 x - 6 ; y2 = 9 - x2,
but onlyy = 2x - 6 is a function.
As:c2+y2 = 9 represents a circle with radius 3 about the origin
of the coordinate system, we can visualize that for every value
of x or y between -3 and 3, there are two values y and x,
respectively, that satisfy the equation. On the other hand, the
explicit forms
y = f(x) = V9-jc2 and y = g (x) = -^9 - x2 ,
representing the upper and lower semicircles, are functions.
Domains and Ranges of Functions of
One Independent Variable
The collection of all values that the independent variable
can assume is the domain of the function; all values taken by
the dependent variable represent the range of the function. A
real-value function of one real variable (that is, one
independent variable) can be represented in a two-dimensional
orthogonal coordinate system, the domain referring to the x-
axis, the range to they-axis.
Find the domain and range of fix) = 1 - x2, where x is a real
number.
Since the only restriction imposed on x is that it must be a real
number, the domain of the function must be the collection of all
real numbers,
]-°°, °°[ •
To find the range of the function, consider the expression
1 -x2 ;
the numerical value of x2 will always be zero or positive. As x
goes toward ooor-oo, f(x) approaches -«>. The function will
reach its greatest value when x is 0. Hence, the range of the
function contains the number 1 and all real numbers less
than 1,
-oo<f(x)<l or J-oo, 1] .

340
Chapter 10 INTRODUCTION TO FUNCTIONS
Find the largest possible domains of
f(x) =
X
g(x) =
x-3
X
x- 3 '
where x represents a real number.
Since division by 0 is not permissible, the domain of/"must be
all real numbers except 3; and the domain of g must be all real
numbers except 0.
x
g(x)
g(x)= x
1 1 1
-40 -20
j
-1
3
r
i
20
40
X
Even and Odd Functions
A function is said to be even if f(-x) = fix) for every value of x
in the domain of definition. The graph of an even function is
symmetrical about the ordinate axis.
A function is odd if f(-x) = - f(x) for every value of x in the
domain of definition; the graph is symmetrical with respect to
the origin.
A
s<j(x,y)
/-*s(-x,y)
—X
(x, y) >^-v
X
X
X
Even Function
Odd Function

Section 10.1 Groundwork
341
Monotonicity, Extrema, Inflection Points
A real-value function is said to be increasing over an interval
if greater values of the independent variable within the
interval produce greater values of the dependent value, and to
be decreasing if the dependent variable decreases with
increasing values of the independent variable. Such intervals
- over which a function either increases or decreases - are
intervals of monotonicity.
Below, the interval ]x\, X2\ is an interval of increasing
monotonicity, while the interval ]x2, x$[ is an interval of decreasing
monotonicity of the function:
Inflection Point
Greek, asymptotos, "not falling
together"; a, "not";
sym, "together";
ptotos, "falling"
(Absolute) Maximum Value
(Absolute) Minimum Value
Relative (Local) Maximum
Value
Relative (Local) Minimum
Value
X
inflection
point
x
A point within an interval of monotonicity where the curve
changes from concavity to convexity, or vice versa, is an
inflection point.
As the independent variable proceeds toward positive or
negative infinity, the corresponding points on the graph of a
function may approach successively nearer to a straight line,
the asymptote.
The value of a function may attain its greatest (inaxiinuin
or absolute inaxiinuin) and lowest (minimum or absolute
minimum) values on an interval either at the endpoints or at
points between the endpoints of the interval. The greatest and
lowest values occurring between the endpoints are also
relative (or local) maxima and relative (or local) minima,
respectively, meaning that they are points where the value of
the function is greater or lower than at all nearby points.
At a relative maximum, an interval of increasing
monotonicity changes into an interval of decreasing monotonicity;
at a relative minimum an interval of decreasing
monotonicity continues into an interval of increasing monotonicity.
A relative maximum or a relative minimum can also be the
absolute maximum or the absolute minimum of a function.
l
maximum
and
relative maximum
relative maximum
relative minimum
■x
relative minimum
minimum

342
Chapter 10 INTRODUCTION TO FUNCTIONS
Extreme Values, Extrema
A collective term for maxima and minima, whether
absolute or relative, is extreme values or extrema (singular:
extremum).
It is common practice to refer to an absolute maximum or a
relative minimum as simply maximum or minimum when it
is evident from the context what kind of extremum one is
dealing with.
Domains and Ranges of Functions of More
than One Independent Variable
The domain of a function of two real variables may be
represented graphically in an orthogonal coordinate plane.
A domain defined by ]-<*> < x < <*>[ and ]-<*> < y < <*>[
corresponds to the whole (x, y)-plane, whereas a domain defined by
]-oo < x < oo[ and ]-oo < y < 0] corresponds only to the lower half-
plane and the jc-axis:
y
A
i
x
x
x y
With the condition — + — < 1, the domain of definition is the
lb y
interior of an ellipse centered at the origin, the major axis of
the ellipse 8 units along the jc-axis, the minor axis 6 units
along the y-axis:
x
To represent graphically both the range and the domain of a
function of two independent variables, we must use a three-
dimensional orthogonal coordinate system. The domain is
still represented in the (x, y)-plane, but may be conceived as an
(x, y, 0)-plane.

Section 10.1 Groundwork
343
Assume a function z = fix, y). The domain of/"may be
represented in the ix, y)-plane; the range off is the collection of
values of z coordinates of all points on the graph. The
collection of all points ix, y, z) that satisfy the equation z = fix, y) is
the graph of f in 3-space.
[x,y,fix,y)\
Calculations with Functions
Functions with a common domain can be combined
algebraically:
fix)+gix) = if + g)ix) fix)-gix) = if-g)ix)
fix)gix) = ifg) ix)
f(X) =(tK*), if£(*)*0
gix)
Iff ix) = x2 and g ix) = x - 1 ,
x
we have

344
Chapter 10 INTRODUCTION TO FUNCTIONS
the sum:
f(x) + g (x) = xl + (x - 1) = XZ + X - 1
_,v2
the product:
f(x)g(x)=x2(x-l) = JC3 -x2
1 I
-5-4-3-2
-5-4-3-2
fa)«to
the difference:
_ ^.2
f(x)-g(x) = xz- (x - 1) = JC2-JC + 1
'(f-*)<*)
-5-4-3-2 -1
the quotient:
X'
fix)
g (x) X - 1
/fr)
I I
-5-4-3-2 <f
■i i i'
U 2 3 4 5
the difference:
g (jc) - /*(jc) = (x — 1) - X1 = -JC2 + JC - 1
-5-4-3-2 -
-■—•»• x
2 3 4 5
<*-/)«
the quotient:
g (x) x - 1
7u7 =^i"
/to)
A
5
4
3
2
1

Section 10.1 Groundwork
345
Composite Functions
COURS DANALYSE
Dt
L'ECOLE ROYALE POLYTECHNIQUE j
P*a M. Acgcstix-Louis CAUCHY,
im f*m»*i CUW«, fntmmt /JU*i*M k TtuUjtljtmkmi^mm,
lapatcar
Ifcata 4« TXtvUm* im wmit, CWtfiar it U UgM*
L" PART IE. ANALYSE ALC&MAJQOE.
DE LIMPRIMERIE ROYALS*
Ck«s Duc&s frcrca, Ubrairu 4u Rot ct 4i U Bibfiokt^uc <lu Roik
im ScrpcaM, a.* 7-
1821.
J. 3. Des Fonctions composed.
Lcs fonctions qui sc deduisent d*une variable &
f aide de plusieurs operations prenneat le aom de
fonctions composees; etfon distiague parini ecsder-
niercs les fonctions de fonctions orui resultent de
plusieurs operations successive*, la premiere
operation ctant eftectuee sur la variable r et cbacune
des autres sur le rcsultat de ropcration precedente.
En vertu de cts definitions >
x9, yx, —, &c....
tont des fonctions composees de la variable x; et
/(tin. x)9 /(cot. x)r &c. .«•
des fonctions de fonctions, dont chacune resuite de
deux operations successive*).
CAUCHY Augustin-Louis
(1789 -1857)
The French mathematician Augustin-Louis Cauchy was a
pioneer who introduced rigor and clarity into modern
mathematics during the 1820s in his research, lectures, and textbooks.
If
y = f(z) = z2 and z = g (x) = 2 x - 1,
where y, z, and jc are variables such that y depends on z and z
depends on x, then y will depend on jc, and we may write
y = (2x-l)2 or (/^)00 = (2^-1)2.
The symbol /*og reads "The composite, or composition, of the
two functions g and f" .
A composite function is a function of a function.
In a function such as (fog)(x), fis the external function andg
the core function.

346 Chapter 10 INTRODUCTION TO FUNCTIONS
In the example below, we show that, generally, (fog)(x) does
not equal (g o f) (x).
Forf(x) = x2 + 3x-2 ; g (x) = x-1, find
(fog)(x)\ (go/)(x); (gog)(x).
(fog)(x) = /-^(^=/-(^-1) = (^-1)2 + 3(^-1)-2
= jc2 + x - 4 .
(go/) (x) = g[f(x)] = g(x2 + 3x-2) = (x2 + 3x-2)-l
= X +OX— O .
(gog)(x) = g(x-l) = (x-1) -1 = x-2.
Inverses
If y = /"(x) is equivalent to x = g (y), the function g is said to be
the inverse of/*, and/"the inverse ofg. Denoting these inverses
by /*_1, g_1, we have
Z*-1 [/(*)]= x; s^feWN y.
One-to-One To have an inverse, the function must be one-to-one, that is, a
functional relationship must also exist when the dependent
and independent values are interchanged. The domain of/*_1
is the range of/*, and the range of f'1 is the domain of f
• Find the inverse (if it exists) ofg (x) = x2 - 2 .
y
= g(x) = x2 -2 ; x2=y + 2; x = ± Vy + 2 ,
which shows that g(x) is not a one-to-one function and,
consequently, has no inverse function.
Find the inverse (if it exists) of
fix) = \x ; x > 0 .
Solve for x and check if y = /*(x) = yx is a one-to-one function:
x = y2 .
Interchanging x and y iny = f(x) = yx and solving for y9 we
have
x = Vy; y = x2,
and /*~ 1 is the function whose domain is [0, «>[, such that
/*_1 (x) = x2 .

Section 10.1 Groundwork
347
Plotting the graphs for f(x) and /*"1(jc) for values x > 0, we
obtain
i
5-
4-
3-
2-
1 -
/A*)"1
/
/
/
/
/
/
/ /¾)
/ ^,.-^-----
/ ^0^-00^
/ _^-*"^
9^00^00^
^ST
ft
L**
i i i i i <
12 3 4 5
X
Similarly:
x + 1
f(x)= —-— <=> f'1 (x) = 5x- 1;
/*(jc) = jc3 <=> /^Oc) = V*
-3 -2 -1
4-
3-
2-
1 -
-
i
r
/
/
/ -
r
/j
/
/
/
/
/
/
1
t
t
1
' 1
-2
-3
-4
ftc)"1
i
2
/w
1 '
3
/Xr)=*»
x*
= V*
1
*(*) = ^(jc)"1 =
A function and its inverse may be equal, e.g.,
g(x) = — = g-1 (x); jc * 0 .
= 1/x x
To verify this, first solve for x and check if y = g(:c) = 1/x is a
one-to-one function: x = 1/y
Interchanging jc and y in y = g(x) = 1/x and solving for y,
we have
x = 1/y; y = 1/x ,
and thus,
g ~1 (*) = — .
► x

348 Chapter 10 INTRODUCTION TO FUNCTIONS
Iteration and Chaos
If a nonlinear function is iterated so that the value from every
iteration is included sequentially into the next one - fix),
f\f(x)], f{f\f(x)]}... - amazing results may be expected.
Let's look at the iteration off(x) = 1 - ex2 with an initial value
(- 1 < x < 1) and c a constant, here called a parameter.
Whatever the initial value of x (within the given interval),
only the value of the parameter - sufficiently small - will
determine the general behavior of the iterated function.
Iterating this function (with the use of a computer) for c = 0 to 2,
the results shown in the diagram to the left are obtained.
For c = 0 to 3/4, every result tends to a stable point; at c = 3/4 we
have a bifurcation point, that is, a point in the parametric space
where the behavior mutates so that every result oscillates
between two values in the dynamical system being studied; at
c = 5/4 the bifurcation doubles and this repeats at higher values
of c, until finally a so-called chaos is reached, where results
jump around in an apparently haphazard pattern that is
dependent on the initial value of x. This is called "sensitive
dependence of behavior dependent on the initial conditions".
The mathematical model of chaos has attracted researchers in
the study of certain predicted changes in population growths
(bacteria, viruses, animals, plants) that failed to occur;
disastrously misleading weather forecasts; sudden major
financial crises, etc. Insufficient data are usually the cause
of miscalculated results of a succession of events in a
dynamical system, but a "sensitive dependence of behavior
dependent on the initial conditions" may also be a reason for a
chaotic behavior that had not been predicted.
Iteration of functions with complex numbers will be discussed
in Chapter 17, "Fractals".
10.2 Elementary Functions
Elementary functions are real-value algebraic functions or
transcendental functions (trigonometric, hyperbolic,
exponential, logarithmic) and all those functions that can be
obtained from these functions through addition, subtraction,
multiplication, division, or by the process of forming
composite functions or taking the inverse of the functions.
Algebraic Functions Algebraic functions are definable in terms of a finite number
of polynomials and roots.
A polynomial function f is defined as
f(x) = aQXn + a\xn~^ + ... +an_\x +an,
where n is a non-negative integer and ao> al> ••• an are real
numbers.
Results
0
3/4
$--;. •'»"■' 0.5
?>'.■>.-.». 0
" }>- 1
5/4 2
Stable Point
Bifurcation Point
Chaos
(Dear Mom + (Dad,
Last month I reached a jvrkjn
the road-a bifurcation-point. My
money oscittated- enough, not
enough, enough, not.... The
bifurcation points have nozu
doubled, lb avoid OiAOS, please
SEO^P MO^E MOO^Erf. Since
my behavior depends sensitively on
the initial amount of money, please
double the amount.
(Withallmylove, ^
•
p. 633

Section 10.2 Elementary Functions
349
Transcendental Functions
A rational function/"is an algebraic function defined as
tt \ P(x}
f(x) = QM'
where P and Q are polynomials and the domain contains all x
for which Q(x) * 0. For instance,
3 x4 - 2 x + 4
rix) = -—
x° + 5 x
is a quotient of two polynomials and, therefore, a rational
function.
Transcendental functions cannot be expressed in terms of a
finite number of polynomials. They include exponential,
logarithmic, trigonometric, inverse trigonometric,
hyperbolic, and inverse hyperbolic functions.
Linear Functions
= 2x-5
*- x
Since the graph of the function y = f(x) = ax + b is a straight
line, the function is called a linear function.
A straight line is determined by any two of its points, for
instance, by its intercepts with the coordinate axes.
The line f(x) = y = 2x-5 intersects they-axis at
y = 2-0-5 = -5
and the jc-axis at
0 = 2jc-5; x = 2.5.
As shown at left, a line is drawn through the points (0,-5) and
(2.5,0).
Proportionality Constant
Direct Proportionality
Two variables y and x are directly proportional if their
functional relationship is given by
y =f(x) =ax ,
where a is the proportionality constant. This is a special case
of a linear function f(x) =a x + b ; b =0.
The graph is a straight line through the origin of the coordinate
system; the proportionality constant a represents the slope of
the line.
The graph below represents the function f(x) = ax for a = - 1/2,
a = 1, and a = 2:
fix)
1 a=2
a=l
a=-l/2
/lx) = ax-

350
Chapter 10 INTRODUCTION TO FUNCTIONS
Power Functions
A function y = f(x) = xn is a power function of degree n.
lin is an even natural number, then f(x) = xn is an even
function and the graph of the function is symmetrical about the
y-axis; curves for different values of n all contain the origin
and the points (-1, 1) and (1, 1).
If n is an odd natural number, then/Xx) = xn is an odd function
with a graph that is symmetrical about the origin; curves for
different values of n all contain the origin and the points
(-1, -1) and (1, 1):
/to = xn
X
/to
A n=5
n=2l\l/n=S
x
Inverse Proportionality
For n = -1, that is, f(x) = a x'1 (a * 0), the curve is a hyperbola,
expressing an inverse proportionality, where a is the
proportionality constant; the asymptotes are the ordinate and the
abscissa:
/to
a
10 i
5
-10
-5
/to = Ax -i
-I 1—^- x
5 10
-5
-10

Section 10.2 Elementary Functions
351
Quadratic Functions
A function
fix) = a x2 + b x + c,
where a, 6, and c are constants and a * 0, is called a quadratic
function.
If b = c = 0, then we have the simplest case of a quadratic
function, f (x) = ax2, a power function of the second degree
whose graph is a parabola with its vertex at the origin of the
orthogonal coordinate system. With a positive coefficient the
parabola opens upward; with a negative coefficient the
parabola opens downward:
fix) = ax
a=l/2
x
a=-l/2
a=-l
The vertex of the graph of a quadratic function represents
the extreme of the function; a positive coefficient a gives an
absolute minimum, a negative a an absolute maximum.
For the same value a, the graph of
f (x) = ax2 + bx + c
is congruent with that of g(x) = ax2, but with its vertex at
b_ _b2\
'2a' C 4a)'
\
c -
fix)
k
p.
g(x) = ax
— rt'**'
fix) = ax2 + bx + c
2a
x

352
Chapter 10 INTRODUCTION TO FUNCTIONS
fix) = 2x2 + 4 x + 6
The vertex of the graph of
is located at
b
= t-jc2 + 4 jc + 6
x = -
2a
= -4; y = c-
4a
= -2.
Since a is positive (1/2), the graph opens upward.
fix) = \x2 + 4 jc + 6 is congruent with gixx) = j U')2 in a
coordinate system whose origin coincides with the position
(-4, -2) of the original system:
fix) = IT*2 + 4 x + 6
/fc)
A
-JC
JC
JC
(-4, -2)
Instead of using the graph of gixx) = -^ 0c')2, we can plot the
graph from values obtained by computing the function for
various values of x.
The vertex of the graph of
fix) = x2-4x + S; -1<jc<5
is located at (2, -1).
Using the information - 1 < x < 5, we may determine the
following points:
X
1
0
1
2
3
4
5
fix) = X2
8
3
0
- 1
0
3
8
-4jc + 3
fix) = x2 - 4x + 3
t—i—r*^* X

Section 10.2 Elementary Functions
353
Exponential Functions
A function f(x) = ax, where a > 0, is an exponential function to
the base a.
All curves of exponential functions pass through the point
(0, 1) of the coordinate plane. For a > 1, the curves rise toward
the right; for a < 1, they fall. The functions f(x) = ax and
g (x) = a~x are symmetrical to each other about they-axis:
forgix) g(x) for/M
a equals: fix) a equals:
The exponential function to the base e of x is denoted
e* or exp x .
Generally, when reference is made to "the exponential
function", it is understood that the function is to the base e.
The curves of all exponential functions contain the point (0, 1);
exponential functions are conveniently compared by
determining the slope of the tangent at that point.
The base of natural logarithms, e, is the base of the exponential
function whose slope at point (0, 1) is exactly 1.
i
slope
equal to 1
-T ►- X
4
6-
4
2
1 JT
/ -9.
f(x) = e>
i i *
12 3

354
Chapter 10 INTRODUCTION TO FUNCTIONS
Logarithmic Functions
Functional Equation
pp. 154 - 56
Logarithmic functions may be defined by their desired
property: to convert multiplication into addition. This is
expressed by a functional equation as follows:
fix-y) = f(x)+f(y\
where f is the function we seek to define and where x, y > 0
range freely. From this functional equation and the
assumption that f is continuous, it follows that fix2) = 2 f(x) and, more
generally, that f(xP) =p fix).
If we know that/* (a) = 1, then f(aP) =p f(a) =p, so that/"must be
the logarithm to the base a, which has the defining property
y = l°&ax if and only if x = a^ .
The logarithm to the base a of x is the exponent p to which a
must be raised to yield the quantity x; thus, these are
equivalent:
x = a? or loga x = p .
Functional equations of logarithms, together with simple
properties of logio, were used by Briggs and others to compute
the early tables of logarithms. For example, from 210 = 1028
& 103, we see that 10 • logio 2 & 3, so that logio 2^3, with the
logarithm being slightly larger than the approximation.
The graphs of the logarithmic and exponential functions
loge x = In x and ex = exp (x) are shown here for the special
base e, which has the unique property of giving each curve a
slope of 1 where it crosses the axis:
In x
x
While y = loga:c always contains the point (1, 0), the slope is 1
here only for the curve y = loge:c. This property makes loge x
and e* useful and is one of the reasons why we call In = loge x
the natural logarithm.
The domain of e* corresponds to the range of the natural
logarithm of x (hue), that is, all real numbers, and the range
of ex corresponds to the domain of hue, that is, all positive real
numbers. If f(x) = e*, then In x is the inverse function off(x).
The domain of e* is ] - «>, «> [; its range, ] 0, °° [.
The domain of In x is ] 0, «> [; its range, ] - °°, °° [.

355
10.3 Continuity and Limits
Definitions
Below, fis a continuous function; g is discontinuous:
x
A function fis continuous at a point x0 if
lim f(x) = f(x0).
x —» xn
Thus:
Continuous at x
o
Discontinuous at x
o
Ax)
M>)
*
*
X
0
A*) =
*- 4
is not defined at x = 4, for division by 0 is not
s2 - 9
permissible. Similarly, f(x) = «- is not defined at x = 3:
LU
-2
-4-
-6

Chapter 10 INTRODUCTION TO FUNCTIONS
The expression
lim fix) = b
x —> a
reads:
"The limit value of the function f ix), as x approaches (tends
toward) a, is 6",
which means, more precisely, that
for every number e > 0 there exists a number 8 > 0 such that
| f (x) - b | < e when 0 < | x - a \ < 5.
We shall not expound on this definition. For the purpose of
this text, an informal or intuitive concept of a limit value is
sufficient.
For
lim. f(x)
x —>a
to exist, fix) must be defined for all x near a.
It is entirely possible, however, that lim fix) exists but fia) is
x —» a
not defined. For instance, if
x* — 4
f w = —>
then fi2) is undefined, since division by 0 is not permitted.
But
ix + 2) ix - 2)
lim fix) = lim = lim (jc + 2) = 4.
*->2 *->2 x- Z x^2
In expressions such as
am - xm ... + a\ - x + a,Q
lim — —
x^0 bn-xn ... + b1- x + b0
all powers of x disappear when x = 0, and the limit value is t- .
&0
If ao = 0 but 6q * 0, we have
am • :cm ... + a\ - x + ao 0
lim — ■ ■— = — = 0,
*_>() on • :cn ... + oi • jc + oo &o
and if 6q = 0 and ao * 0,
_. am - Xm ... + CL\ - X + ar\ CL\ - X + CLq
lim -— —= lim —
x^0 on- xn ... + bi- x + b0 x^0 oi-x
does not exist, since —■ takes on values which are
b\ - x
increasingly large in size; these values will have one sign for
x > 0 and the opposite sign if x < 0.
The lowest power of x will always decide the limit value, as
x ->0
|a\x | > |«2^21 > • • • > \amxTn\ y
and similarly,
\bix\ > |&2*21 > ••• > \bn xn\ •

Section 10.3 Continuity and Limits 357
jc3 — 8 x
To determine lim
x^0 5 x2 + 4 x
we have, as x —> 0, jc3 « 8 x and 5 x2 « 4 x, and thus
jc3 - 8 x _. - 8 x
lim — = lim — = -2.
x_»o5ar + 4x * _> o 4 *
Similarly,
_ . X ~\~ O X - . o 3C o
lim = lim -— = — .
x^09x2 + 4x x^o4x 4
In expressions of the kind
_. am • xm ... + a\ • x + a§
lim — ; ; ,
*->«, bn' xn ... + bi- x + bo
where x tends toward infinity, we can distinguish three cases:
m>n; m=n; m<n
7jc4-2jc _. 7^4 ,. 7jc
m > n lim = lim —- = lim -r- = oo;
X ^ oo D X <J X x ^ oo \J X x ^ oo
7jc3-2jc ,. 7jc3 7
m = n lim = lim —- = -r- ;
x ^ oo O 3C — O 31 J£ ^ oo O X
.. 7x2-2x ,. 7x2 ,. 7
m < n lim — = lim —- = lim —— = 0.
X —> oo O X O X x —> oo O 3C x —> oo
As for jc —> oo?
7 jc4 » 2 x ; 7 jc3 » 2 jc ; 7 jc2 » 2 jc ; 6 jc3 » 5 jc .
Evaluate
,. Vl + 8x-l
lim .
x->0 *
Since division by 0 is not permissible, it is necessary to
rearrange the given expression so that x is not a factor of the
denominator.
,. VTT8~*-1 ,. (Vl + 8x-l)(Vl + 8* + l)
lim = lim , r
*->0 x *->0 x (Vl + 8x + l)

358
Chapter 10 INTRODUCTION TO FUNCTIONS
Right-Hand Continuity
Left-Hand Continuity
One-Sided Limits
On the real number line, the independent variable can
approach a fixed value either from the right or from the left,
which may be denoted x —» a+ and x —> a_, respectively.
If x infix) = — tends to 0 through positive values,
fix) = lim — = oo,
x-*0+ x
which reads: "The limit value of fix) as x tends to 0 from the
right is infinity."
If x tends to 0 through negative values,
fix) = lim — = -oo9
*->o_ x
which reads: "The limit value of fix) as x tends to 0 from the
left is minus infinity."
A function /"has right-hand continuity at a point x = a if
lim fix) = fia)
x —> a +
and left-hand continuity if
lim fix) = fia) .
x —> a _
To examine a function for continuity at a specified point, it is
sometimes necessary to determine the one-sided values at that
point, fix) is continuous at a point x = a if
lim fix) = lim f ix) = f (a).
x —> a + x —> a _
To draw the graph and assess continuity of
fix) = eyx; x*0
several values of x are tested with the help of an electronic
calculator:
flx) = e17*
i i—p
2
X
4.5
4
3
25
2
1.7
15
1.3
11
1
0.9
0.7
0.6
0.4
0.3
0.2
0.1
0.01
0.001
el/x
1.24...
1.28...
1.39...
1.49...
1.64...
l«oO...
X •X^T* * •
2.15...
2.48...
2.71...
3.03...
5.17...
0^£7. ..
12.1...
28.0...
1.4... x 102
2.2... x 104
2.6... x 1043
1.9... x 10434
JC
-4.5
-4
-3
-2.5
-2
-1.7
-1.5
-1.3
-1.1
-1
-0.9
-0.7
-0.6
-0.4
-0.3
-0.2
-0.1
-0.01
-0.001
el/x
0.80...
0.77...
0.71...
0.67...
0.60...
0.55...
0.51...
0.46...
0.40...
0.36...
0.32...
0.23...
0.18...
0.08...
0.03...
0.67... xlO"2
0.45... xl(T4
3.7... xKT44
5.0... xl(T435

Section 10.3 Continuity and Limits
359
x
-3^+X + l
The function fails to be defined at a point where x = 0.
Testing the one-sided limit values as x tends to 0 through
positive values, we find that fix) tends to positive infinity:
x—>0+, eyx —»°o .
On the other hand, as x tends to 0 through negative values,
f(x) tends to 0:
x->0_, eVx ->0.
Assess the continuity of
M-i?;"1'
x < 2
x> 2
at
lim
x->2_
x = 2.
fix)
lim i-x2 + x+l) = -4 + 2 + 1
lim fix) = lim i-x + 1) = -2 + 1=-1.
x-+2 + x-*2 +
Moreover,/* (2) = -2 + 1=-1.
= -1
Thus,
f is continuous at x = 2
gix)
-2 -1
xA
1 2
*- x
Assess the continuity of
gix) =
{
X'
2 < x < -1
-1 < jc < 1
at jc = - 1
lim g ix)
x -»-1 _
lim g (jc)
x -> -1 +
•
= lim 2 =
X -> -1 _
= lim jc2 =
JC ->-l +
= 2
(-1)2 = 1
Thus,
g is discontinuous at x = - 1
Find a coefficient a for which
5 - x ;
hix) =
Cf'^V «v
2.
JC < 1
jc< 1
is continuous at jc = 1
5x-jc2
-3 -1 j 2
- JC
lim h ix) = lim (5-jc) = 5-1 = 4.
x-»l_ *-> 1_
lim /i 6c) = lim iax-x2) = a-1.
X —» 1 + JC —> 1 +
Continuity at jc = 1 exists if lim hix) = lim /i(jc) = h (1), that
*—»l_ x-> l +
is, if
Thus,
a- 1 = 4 ; a = 5 .
/i is continuous for a = 5.

360
Chapter 10 INTRODUCTION TO FUNCTIONS
Calculations with Limits
The following rules of calculation with limits apply, provided
that the limits exist for f and g:
lim [f(x) ± g(x)] =
x —> a
lim [fix) • g (x)]
x —> a
lim [q • f(x)\
x —> a
lim
x —> a
'f(x)'
g (x)
lim [f(x)n]
x —> a
lim [f(x)1/n]
x —> a
lim q
x —> a
fix)
lim /* \g(x)]
x —> a
lim [log^/'U)]
x —> a
lim f(x) ± lim # (x)
x —> a * —» a
lim /*(jc) • lim g (x)
x —> a x —» a
= 9
[lim A*)]
L?c -> a J
lim /*(jc)
x —> a
lim g (jc)
x —» a
, if lim g(x) * 0
x —» a
= 9
= /
[lim f(x)ln
\_x -> a J
[lim /•(x)]17*
= logc
[lim g(x)l if/is continuous
L*">a J at 6= limg(x)
x —> a
\lim f(x)~\
Indeterminate Forms
A quotient
lim
x —> a
fix)
gix) '
where fix) and gix) both approach 0 or + °° as x tends to a, is
called an indeterminate form,
0 °°
o or Z-
Indeterminate indicates that such limits may or may not
exist.
Indeterminate forms also occur as products, differences, and
powers.
A product lim f(x) • g(x), where
x —>a
lim f(x) = 0 ; lim g (jc) = °o ,
x —> a x —>a
gives rise to the indeterminate form 0 • », and may be rewritten

Section 10.3 Continuity and Limits 361
T fix) g (X)
lim t,—tt or lim
_ l/g(x) Vf(x) '
0 oo
that is, as an indeterminate form — or — .
U oo
A difference lim [f(x) - g (x)], that is, an indeterminate
x —> a
form (oo _ oo)? may be rendered as
_. l/g(x) - l/f(x)
1 1 '
x —> a ___ . .^-_
fix) g(x)
that is, as a form — .
A power expression lim f(x)8^ may be transformed into
x —> a
lim e^-m/W ,
jc —> a
where the exponent has the form 0 • oo .
The products
0 • In 0 = 0 • (-oo) "
0 • In oo = 0 • oo )>
°° • In 1 = 00 • 0
>
and, consequently, the power expressions 0°, °o°? and I00 are
also indeterminate.
Summing up, the following expressions are indeterminate:
0 00 „ „
- — O-oo 00-00 0° 00O l°°
OO
Many limits of an indeterminate form can be evaluated by a
p. 782 formula, L'Hospital's rule, which uses methods of differential
calculus.
Determinate Forms
The forms
00 . 00 = oo* 00 + 00 = 00* — 00 — 00 = — 00
are evidently determinate, in the sense that, for instance, if
lim f(x) = lim g(x) = 0,
x —> a x —> a
then
lim f{x) • g(x) = 0 .
x —> a
Other determinate forms are
ni—
000; v^.

362
Chapter 10 INTRODUCTION TO FUNCTIONS
Undefined Forms
Expressions of the form
a
—, where a is a non-zero real number, or °o
a
are undefined, because if y is very small, then — will be very
large in size but positive or negative according to the sign of y
and a.
Also,
because
but
0~°° is undefined,
1 i m x~1/x = + oo,
x —> 0+
lim x~s (x) _ _ oo? if g (^) equals, for instance, the greatest odd
*-> 0_
integer <
x
Determine the limit (if it exists) of
x2 + 3 x + 2
lim
*_>() 9xz + 4x
If
then
fix) =
x2 + 3 x + 2
9 x2 + 4 x
lim f{x) = <*> ; lim /"(jc) = - °° ,
x -> 0+ x -> 0_
where neither statement says that the limit exists and has the
value oo (or - <*>), but rather each statement says that the limit
fails to exist in a particular way.
© STORM P.- MUSEUM, COPENHAGEN

363
Chapter
11
OVERTURE TO THE GEOMETRIES
(De 'Matificis et Mathematicis et Ceteris SimiCiSus
Sirtem geometriae discere atque e?(erceri publice intersit, ars autem
mathematica dammabUis interdicta est, ... Haruspe^ ... qui kuic
ritui adsoCent ministrare ... concremando ifto Haruspice ...
Corpus Juris Civilis, Codex Justinianus,
Book IX XVIII, 2,3 (AD circa 650)
Or, in other words:
Concerning Mathematicians and Soothsayers and
Kindred Evildoers
The study and teaching of the science of geometry are in the
public interest, but whosoever practices the damnable art of
mathematical divination, shall be put to the stake.
Read on at your own peril!
Page
11.0 History 364
11.1 Geometric Abstraction 370
11.2 Perspective and Projection 372
11.3 Form and Shape 376
11.4 Survey of Geometries 377
11.5 Topology 378
11.6 Euclidean and Non-Euclidean Geometries 381

364
Chapter 11 OVERTURE TO THE GEOMETRIES
11.0 History
Greek, ge, "Earth";
metria, "measurement"
The oldest suggestions of an ordered system of
measurements go back to the ancient Babylonians, who developed
methods of land surveying embodying calculations of the area
of simple geometric figures bounded by straight lines and arcs
of circles. This is reflected in the name "geometry", whose
literal meaning is Earth measuring.
The assumption of Babylonian astronomers that the year had
360 days is very likely the origin of our system of measuring
angles in degrees; the fact that the angles of equilateral
triangles are 60° may explain, in part, the sexagesimal
method of counting. The Babylonians laid the first
foundations of the science of geometry by their purposeful study
of the properties of circles.
Unlike the Egyptians, whose interest in geometry lay
exclusively in the practical considerations of land measurement,
the Greeks devoted their energies to a systematic study of
geometrical figures and their properties to establish a new
science.
r\ V^k
PLATON
(427 - 348 or 347 B.C.; original
name ARISTOCLES; at school
given the nickname Platon in
reference to his broad shoulders;
Greek, platon, "broad")
Greatest among ancient Greek scientists were Plato and
his pupil Aristotle. In an olive grove, or park, near Athens,
Plato conducted a school of philosophy, Achademya, where
he is said to have erected an ornamental gateway with a
notice proclaiming "No Admittance" for those who knew no
geometry.

Section 11.0 History
365
ARISTOTELES
(384 - 322 B.C.)
EVDOXOS
(c. 408 - 355 B.C.)
Aristotle belonged to those in the know; eventually the most
famous polymath of his time, his teachings came to be
regarded as absolute and inviolable truths, which made them
lasting obstacles to scientific progress for more than 1500
years, all through the Middle Ages.
Though no original works of Eudoxus - a pupil of Plato's -
have survived, we know of them through references made by
Euclid and Archimedes. Eudoxus is remembered for the
theory of proportions, described in Euclid's Elements, and is
usually credited with the invention of the method of exhaustion
to find approximate areas and volumes of curvilinear forms
and shapes - a forerunner of integral calculus.
Together with Eudoxus, the greatest mathematicians of
antiquity, towering over all their contemporaries, were Euclid
and Archimedes - the former in Alexandria and the latter at
Syracuse in Sicily, both cities to be considered culturally as
Grecian.
Title page of the first
complete publication in English
of Euclid's Elements (1570).
Euclid (of Alexandria) is
incorrectly referred to as
"Euclide of Megara", also a
geometer, and a
contemporary of the rightful author
of Elements.
mm
HUSS"*-**
**■**■*!—r+ K.I —™— i

366
Chapter 11 OVERTURE TO THE GEOMETRIES
EVCLEIDIS
(c. 330 - c. 275 B.C.)
Elementary Geometry
ARCHIMEDES
(287 - 212 B.C.)
APOLLONIOS
(c. 260 - after 200 B.C.)
HIPPARCHOS
(? - after 127 B.C.)
PTOLEMAEUS Claudius
(Roman citizen and, therefore,
his name in Latinized form;
c. A.D. 100 - c. 170)
PAPPOS of Alexandria
(c. 300-c. 350)
PROCLUS
(412-485)
HYPATIA
(370 - 415)
Euclid is best known for his 13-book treatise Stoicheia
(ZTOix&a, "Elements"), a collection of all the geometric
knowledge of his day, often referred to as elementary
geometry after the name of the book.
Euclid's problems are all solved by logical reasoning from a
central core of postulates, or axioms. This is the classical
axiomatic method of Euclid. All his problems, solutions, and
proofs rely on straight lines and circles, and his constructions
are carried out using only an unmarked ruler and a pair of
compasses. Euclid concerned himself with problems of plane
geometry, that is, with geometric figures constructed on a
plane surface, and with polyhedra, that is, solids bounded by
plane polygonal regions.
Solid geometry was the province of the mathematician and
scientist-inventor Archimedes, whose contributions to
geometrical science overshadow those of any other person. He
studied, in depth, the properties of the sphere, cylinder, and
cone, and the relations between them, which brought him very
close to the foundations of calculus; he systematized his
findings after the manner of Euclid, and his books are still
useful.
By the 3rd century B.C., Greek mathematician Apollonius
of Perga studied and named the conic sections. Like
Archimedes, Apollonius used longitude, latitude, and altitude
to define the position of a point.
Hipparchus of Nicea and Rhodes is usually credited with
inventing trigonometry, using trigonometric methods in his
calculations of the distances to astronomic objects.
Originally, functions of chords of isosceles triangles were
tabulated for use in spherical trigonometry, in response to the
need for accurate information in astronomy. Today,
trigonometric functions have been generalized to apply to any type of
triangle, and are widely used to simplify calculations in
several other areas of mathematics.
The geographer, mathematician, and astronomer Ptolemy of
Alexandria is mainly remembered for his work on projective
geometry, which came to be the foundation of descriptive
geometry, and for his 13-book treatise Almagest dealing with
the application of geometry to astronomy, which became the
standard textbook for over a thousand years, until Copernicus
published De revolutionibus.
Pappus of Alexandria calculated the surfaces and volumes of
solids of revolution, particularly those generated by the
rotation of conic sections. After Pappus, Alexandria produced
no geometers of note, but a native Greek, Proclus, wrote a
history of geometry summing up the work of Alexandrian
geometers. After Proclus, the history of the University of
Alexandria offers nothing of interest to chroniclers of
geometry. Oppression by Christians and other warring
religious factions caused the university and the science
library to decline toward the end of the 3rd century.
Hypatia, head of the Neoplatonist school of philosophy at
Alexandria and one of the most learned and eloquent teachers

Section 11.0 History
367
ALBERTI Leone
(1404-1472)
DESARGUES Girard
(1591-1661)
PONCELET Jean Victor
(1788 -1867)
MONGE Gaspard
(1746 -1818)
LAMBERT Johann Heinrich
(1728-1777)
DESCARTES Rene
(1596-1650)
of antiquity, wrote commentaries on Apollonius's Conies and
on several other works in mathematics, all of them now
vanished. She was barbarously murdered by a mob of so-called
Christians who equated science and learning with paganism.
The murder of Hypatia was followed by an exodus from
Alexandria of many Greek scientists and
mathematicians who settled in Persia and in Arab countries - mainly in
the city of Baghdad - where they became teachers. Many
scientific books from the library were saved from destruction,
among them Euclid's Elements and Ptolemy's Almagest,
which were translated into Arabic and eventually into Latin
and became standard works in mathematics, geometry,
trigonometry, and astronomy.
The Arabs carefully tended and cultivated the knowledge that
they acquired from the Greek scientists of Alexandria. Some
of the world's most famous and learned scholars of the six
centuries following the fall of Alexandria, in A.D. 641, were
Arabs.
Arab scientists were very successful in applying their
acquired knowledge of geometry and trigonometry to
astronomy. They also made an important contribution to the
history of geometry - and to science in general - by the
preservation of the scientific knowledge of antiquity through
the Dark Ages of European science. They translated learned
works of science, and it is through this work that, in the 12th
century, the science of geometry finally found its way into
Europe via Moorish Spain.
The Italian architect Leone Alberti was the first who, when
discussing perspective in art, formulated ideas that pointed
the way toward a future science of projective geometry. Girard
Desargues, in his turn, was the first professional
mathematician who, exploring the logical foundations of Euclidean
geometry, initiated a formal study which extended Euclid's
methods to projective geometry. Jean Victor Poncelet in 1822
published a treatise which revitalized interest in the subject of
projective geometry as a science that treats form and position
distinct from size.
Descriptive geometry, in itself a form of projective geometry,
may be said to have been created in 1794 when Gaspard Monge
published his Geometrie descriptive. It takes up many
features already known from works by Desargues and
Lambert, but the credit for having developed projective
geometry into a science in its own right belongs to Monge.
Rene Descartes published, in 1637, La geometrie, showing how
geometric figures could be analyzed algebraically. Analytic
geometry - the geometry where positions are represented in a
coordinate system and their properties analyzed algebraically
- evolved under the influence of Descartes's work, but La
geometrie itself does not use "Cartesian" coordinate axes or
any other coordinate system; it is as much concerned with the
rendition of algebra into terms of geometry as vice versa. As

368
Chapter 11 OVERTURE TO THE GEOMETRIES
understood by Descartes, every algebraic step in an argument
had to correspond to a geometrical construction. A remarkable
feature of the text is that Descartes departs from the Greek
practice of regarding a2 and a3 as area and volume; he
represented them as lines.
La geometrie is the earliest mathematical text that a modern
student of mathematics could read without stumbling over an
abundance of obsolete notations.
DISCOURS
DE LA METHODE
Pour bien conduire fa rai(bn,& chercher
la verite danslesfciences.
Plus
L A DIOPTRIQVE.
LES METEORES.
ET
LA GEOMETRIE.
Qui font desejfais de cete Methode.
L A
*>7
A L E Y D E
Dc rimprimerie de L a n Maire.
cla la C xxxvli.
Auec 'Priutlege.
ap8 La Geometrie.
eft a 1'autre, ce qui eft le mefme que la Diuifion; ou enfin
trouuervne,ou deux,on plufieurs moyennes proportion-
nellesentrervmte*, &quelque autre ligne j ce qui eft le
mefme que tirer la facine quarre*e, on cubique,&c. £t ie
ne craindray pas d'inttoduire ces termes d'Arithmeti-
que en la Geometrie , affin de me rendre plus intel-
ligibile.
Soit par exemple
ABl'vnite*. &qu'il
faille multiplier BD par
B C, ie n'ay qu'aioindre
les poins A & C, pais
tirer D E parallele a C A,
&B£eft leproduitde
cete Multiplication.
Oubien s'il faut diuifer B E par B D, ayant ioint les.
poins E & D, ie tire A C parallele aDE.&BC eftle
produit decete diuifion.
Ou s'il faut tirer la racine
qaarree de GH, ie luy ad-
ioufte en ligne droite F G,
qui eft l'vmtcf, & diuifant F H
3h en deux parties efgales aa
point K, du centre K ie tire
le cercle FIH, puis efleuant du point G vne ligne droite
iniques a I, a angles droits fur F H, e'eft GI la racine
cherche'e. Ie ne dis rien icy de la racine cubique, ny des
autres,acanfequei'enparleray plus commodement cy
apres.
Mais foauent onn'a pas befoin de tracer ainfi ces
ligne
GEOMETRIE.
LIVRE PREMIER.
<Des prohkfines cpton peut conftrusre fans
y employer que des cercies & &*
(tgnes drones*
&£38§kOus les Problefmes de Geomerrie fe
I pcuuenr fecilemenr reduire a rels rermes,
j qu'il n eft befoin par aprds que de Connoi-
. ftre la longeur de quelqaes lignes droires,
I poor les conftruire.
Er comme roure I'Anrhmerique n'eft compose, que
dequatre ou Cinqoperarions. qui fbnr 1'Addition. la
Sooftradhon, la Multiplicarion, la Dinifion, & I'Extra-
&ion des racines. qu'on pcur prendre pour vac efpece
de Diuifion t Ainfi n'aronaurrechofe a faire en
Geometrie ronchanr les lignes qu'on cherche. pour les
preparer a eftre Connue's. que leur en adioufter d'autres. ou
en ofter. Oubien en ayanr vac. que ie nommeray 1'vnird
pourlarapporterd'anranrmieuxauxnombres , &qui
peurordinairemenr eftre pnfe a difcrerion.pws en ayant
encore denx autres. en trouuer vne quatriefine, qui (bit
al'vne de ces deux,comme rautre eft a rvnite\ ce qui eft
lemefmequelaMultipUcationj oubien en trouuervne
quatriefincquifoitalVnedecesdeux, comme IVmte*
Pp eft
LivRE Premier. i9f
goes fur le papier, & il fuffift de les defigner par quelques
tectrcs, chafcune par vne feule. Comme pour adioufter
la ligne B D a G H, ie nomme 1'vne a & 1 autre b,&c efcris
a •+- b; Et a - 6,pour fbuftraire b d' *t Et a 6,pour les
multiplier 1'vueparfautrejEt ~, pour diuifer 0 par4 ;Et aa,
ou at pour multiplier a par foy mefine s Eta, pour le
multiplier encore vne fois par a, & ainfi a 1'infini . Ec
•~l T • a
Y a-hb, pour tirer la racine quarrcc d' a •+• b; Ec
^ C. a— b -i-a bb, pour tirer la racine cubique d'a—b
•+• abb, & ainfi des autres.
Ouilcfta remarquer que par a ou b ou femblables,
ieneconcpyordinairement que des lignes toutes fim-
pies, encore que pour me feruir des noms vfites en f Al-
gebre, ie les nomme des quarrel ou des Cubes, 3cc.
Jleftaufly a remarquer que toutes les parties d'vne
mefine ligoe/e doiuenr ordinairement dp rimer par aa*
tant de dimeniions IVne que 1'autre. lorfque I'vniteVeft
point ddcerminee en la queftion, comme icy a en con-
tientautancqu'066 ou b dont fecompofe la ligne que
l'ay nomme*e ^C. a - b -t- a bb: raais que ce n'eft
pas de mefine lorfque I'vnite' eft ddtermine'e, a caufe*
qu'ellc peut eftrefbufcntendue par tout ou il y a trop on
troppcude dimeniions: comme s'il faut tirer la racine
cubique deaabb — b , il faut penfer que la quantity
aabbeA diuifee vne fois par IVnicc', & que 1'autrequan-
tire* b eft mulcipliee deux fois par la mefme.
Pp a Au

Section 11.0 History
369
SACCHERI Girolamo
(1667 -1733)
LOBACHEVSKI Nicolai
(1792 -1856)
BOLYAI Janos
(1802 -1860)
GAUSS Carl Friedrich
(1777 -1855)
RIEMANN Bernhard
(1826 -1866)
Greek, topos, "place"
MOBIUS Augustus Ferdinand
(1790 -1868)
LISTING J. B.
(1808 -1882)
BETTI Enrico
(1823 -1892)
JORDAN Camille
(1838 -1922)
POINCARE Jules Henri
(1854-1912)
HILBERT David
(1862 -1943)
Greek, homos, "same";
morphe, "shape", "form"
Topological Invariant
After the discovery that geometric problems could be
transformed into algebraic problems, mathematicians, around
1700, began to apply calculus to the study of geometrical curves
and surfaces. This eventually led to a new branch of geometry
known as differential geometry, now of paramount
importance in science.
First to question the validity of the Euclidean parallel axiom
was the Italian mathematician Girolamo Saccheri; his ideas
were forgotten, however, until they were rediscovered in 1889
by his compatriot Beltrami. Another early attempt to devise
a new geometry was made by Lambert in his Theorie der
Parallellinien, published posthumously in 1786.
Other mathematicians persisted in developing non-Euclidean
geometries. Nicolai Lobachevski in 1829 and Janos Bolyai in
1832 independently described hyperbolic geometry; here the
ubiquitous Gauss also cast his shadow. In 1854, Bernhard
Riemann presented elliptic geometry - so named in 1871 by
Felix Klein - and general Riemannian geometry, which,
incidentally, is the mathematical foundation of Albert
Einstein's general theory of relativity.
While the branches of geometry mentioned above depend on
measurement of length and angle and therefore are referred to
as metric, topology is a non-metric geometry.
An offshoot of 17th- and 18th-century findings by Descartes
and Euler, correlating the number of vertices, edges, and
faces of polyhedra, topology is mainly an invention of the late
19th century, initiated when one-sided surfaces were
discovered, independently, by the astronomers Augustus Mobius
and J. B. Listing. In 1848 Listing published, at Gbttingen, Vor-
studien zur Topologie ("Introductory Studies in Topology"),
thereby introducing the term "topology" into mathematics.
Other pioneers in topology were Bernhard Riemann, Enrico
Betti, Camille Jordan, and Jules Henri Poincar£ - "the last
mathematical universalist", who in Analysis situs (1895)
launched the foundations for the development of topology.
A dominant figure in modern geometry was the German
mathematician David Hilbert, professor at the University of
Gbttingen from 1895 until his death. In Grundlagen der
Geometrie (1899; "Foundations of Geometry"), Hilbert gave a
logical examination of Euclidean geometry, and in 1900, at an
international congress of mathematics in Paris, he submitted
23 important mathematical problems to be targeted during the
20th century; many of Hilbert's problems have been solved.
Solved or unsolved, these problems have been and remain an
enormous stimulation for the general development of
mathematics.
Objects are topologically equivalent, or homeomorphic, if one
object can be transformed into another by topological
transformation, that is, by bending, stretching, or twisting, but not
by overlapping, tearing, or cutting. A topological invariant is
a property held in common by all homeomorphic objects.
Topology today is not limited to classical geometrical
configurations but is used in connection with invariant properties

370 Chapter 11 OVERTURE TO THE GEOMETRIES
of equations and functions. It has generalized mathematical
application to the study of continuity.
The German mathematician Felix Klein presented in 1872, as
an inaugural lecture for a chair of mathematics at the
University of Erlangen, his celebrated Erlangen program, a
systematization - from the standpoint of invariants - of all
the then known fields of geometry; this systematization was
based on his and the Norwegian mathematician Sophus Lie's
expansion of the group theory to include geometry. After
Erlangen, Klein attained the chair of mathematics at the
University of Gdttingen, made famous by Gauss. Besides his
pioneering role in many fields of mathematical research,
Klein was a leader of reforms of education; he introduced the
heuristic method in mathematical education, that is, the use of
a process of reasoning that, without pretensions to rigor, often
leads to a correct result.
David Hilbert's work on invariants between 1890 and 1893 and
the Dutch mathematician L.E.J. Brouwer's 1911 work on
topological invariants are important landmarks of modern
topology, whose foundations had been laid by Poincare.
Greatly influential in modern topological research and
teaching were the American Oswald Veblen and the
Englishman Henry Whitehead, whose many joint publications
include the Foundations of Differential Geometry (1932), a
classic in modern geometric research.
The 20th century has seen an increasing role of geometry in
science, from relativity theory and the differential geometry
of fundamental particles and fields to catastrophe theory in
biology and psychology and the geometry of complex
dynamical systems, chaos, and fractals. These applications
have led to new and surprising developments in pure
mathematics.
11.1 Geometric Abstraction
Qeometry exists as an innate -phenomenon in our consciousness.
Peter H0eg, Smilla's Sense of Snow (1993)
A purely spatial abstraction of the visible world in three
dimensions - length, breadth, and height - generally
presupposes that color, surface texture, and other essential
qualities be ignored.
Many of us have taken a circle of string, wrapped it around our
hands, and asked a friend, "Can you take this?" The game, of
course, is cat's cradle.
Around the world, children and adults can be seen playing
cat's cradle and making other string models. In the
KLEIN Felix
(1849-1925)
Erlangen Program
LIE Marius Sophus
(1842 -1899)
Galois theory: p.: 301
Greek, heuriskein, "to discover"
cf. Eureka!, "I have found it!"
BROUWER
Luitzen Egbertus Jan
(1881 -1966)
VEBLEN Oswald
(1880 -1960)
WHITEHEAD
John Henry Constantine
(1904-1960)

Section 11.1 Geometric Abstraction
371
industrialized world it is mostly a game, but in those isolated
societies where cat's cradle has retained its traditional
significance, this kind of string weaving is sometimes used to
teach about animals, leaves, and other objects, or to illustrate
the narrative of a legend or a recent hunt.
Australian Aboriginal open-air classroom. The pattern of a turtle is
formed with bark twine. Photo: Belinda Wright © National Geographic Society
String model of seal (Greenland).
Photo: Ivars Silis © National Geographic Society
It is noteworthy that such abstract models of nature are used
and understood by people all over our planet; identical figures
of string weaving may be presented by people as far apart
as the Batwa pygmies of Africa, the Navajos, Apaches,
Cherokees, and Eskimos of North America, and the tribes
of Borneo, the Philippines, New Guinea, New Zealand,
Australia, and Hawaii.
Pick up a piece of string, make a triangle, a square, a
rectangle, a parallelogram, a trapezoid ... or a temporary work of
art!

372
Chapter 11 OVERTURE TO THE GEOMETRIES
11.2 Perspective and Projection
Focused Perspective
Latin, perspicere, "to look through'
Developing a system of focused perspective, painters of the
Renaissance departed from the traditional lack of depth and
created a visual geometry as opposed to the prevailing tactile
geometry. In a focused perspective, objects are depicted as they
appear to the eye - the objects shrink with distance and
parallel lines seem to converge to vanishing points.
pafpmiua'
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rfcf * Sfafyfdtafcf ota 2mial*3uMftJ) aflat funfllictyatarit' ftirannlubfrcu
^flalcnt/^filtyatrcra (MtyVfcmtai;CnDoifhcfan (Ztnwnami/
^cfrtaiKTa/audjaHroattKTO/iHAtoto^ 'V<r-
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aucfcfWAcfuntflaclkcT/ frail autfcriidxn fcrnio^mui^
ten Wcfrni/ be jroffai n* icrnat mag / nut vie 114^
Kit carju MatatOcn fouix*
ROLLER Hieronymus
(1539 - ?)
Title page of Hieronymus Roller's Perspectiva, a textbook for the
craftsman; published in Frankfurt in 1546.

Section 11.2 Perspective and Projection
373
WMI^mmW*^^
Projection
The fundamental principle of focused perspective is the
projection of the object on the plane.
The perpendicular AA' is at right angles to every straight line
in plane P that passes through the point of intersection between
the plane and the perpendicular. The point of intersection, A'
- called the foot of the perpendicular - is the projection of point
A onto plane P. A'B' is the projection of the distance AB.
A'BC is the projection of all points on the perimeter of ABC,
illustrating perpendicular projection, just one limited
application of projection. The Albrecht Diirer (1471 - 1528)
woodcut below suggests another and more intuitive projection.
mm*mmmm*^a—******i*i**i**4mm*mim*^^—m*+im4rr*mmmmimmmmmm
STOR Lorenz
(?-c. 1621)
German woodcarver and
painter; author and
illustrator of Geometria et
Perspectiua (1556).

374
Chapter 11 OVERTURE TO THE GEOMETRIES
Phantasmagoric Geometries
REUTERSVARD Oscar
(b. 1915)
PENROSE Lionel Sharpies
(b. 1898)
PENROSE Roger
(b. 1931)
"Tribar"
(L.S. and R. Penrose, 1958)
The Swedish artist and art historian Oscar Reutersvard's 1934
drawing of an "impossible triangle" (Opus 1), and the many
works that followed, opened an entire world of undecidable
figures and shapes never before imagined. Reutersvard uses
a "Japanese perspective", where all parallel lines remain
parallel and do not meet at points of visual convergence.
A revival of interest in this art form and a wider interest in
Reutersvard's work came in 1958 after the article "Impossible
Objects: A Special Type of Visual Illusion", published in The
British Journal of Psychology, by the English geneticist L. S.
Penrose and his son Roger, a mathematician, later known for
his work, with Stephen Hawking, on black holes in space,
and for the discovery of aperiodic tilings; the article was
illustrated with a "tribar".
Reutersvard's explorations of a world of the impossible, or
undecidable, have gained the attention not only of artists but
scientists as well, primarily for the study of visual perception
in psychology.
"Opus 1" (1934)
Drawings by Oscar Reutersvard

Section 11.2 Perspective and Projection
375
No presentation seems needed here of the paradoxical visual
ESCHER Maurits Comelis and perspective effects of the images of Dutch graphic artist
(1898-1970) M. C. Escher, whose art is widely reproduced.
Op Art Works of optical art, or op art, which emerged in the 1950s, are
creations of illusory movement, devised by simple repetitive
configurations with gradual, small changes of form and
pattern or by "chromatic tension" from the juxtaposition of
complementary colors.
Oiow to Catch TLleiphants - an 'Elephantasy
If you want to catch elephants, this is how you do it according to
the Copenhagen newspaper Berlingske Tidende.
Sill you need is a blackboard, a piece of chatty a mariner's telescope,
a pair of tweezers, and an empty jampot,
you begin by writing 1 + 1 = 3 on the blackboardand set it up in a
place rich in elephants, and hide yourself in a nearby tree.
Very soon an inquisitive little baby elephant will draw near to
muse upon this remarkable statement, and by and by several of her
elders will join her.
When there are enough of them, you just turn your telescope the
wrong way around, so that the assembled elephants become quite
small Uhen you pick the small elephants up with your tweezers,
one by one, and put them in the jampot

376 Chapter 11 OVERTURE TO THE GEOMETRIES
11.3 Form and Shape
Plane and Space
In geometry, a straight line has length only, and is of one
dimension; a geometrical shape that has area, but no volume, is
of two dimensions - length and breadth (width); a volume has
three dimensions. A three-dimensional region is called space
or 3-space.
A point - which may be regarded as the intersection of two
lines - has the dimension zero.
The adjective planar refers to a two-dimensional quality in
the plane. Surfaces are always two-dimensional but only
rarely planar ("flat").
Plane geometry is concerned with the properties of plane
figures - geometrical shapes of two dimensions: angles,
triangles, squares, higher polygons, conic sections, etc.
Solid geometry deals with shapes of three dimensions -
figures in space - such as pyramids, prisms, etc., and angles
between planes. Plane sections of solids belong to plane
geometry but may be studied profitably in space.
Euclidean geometry is based on the assumptions of Euclid,
who, about 300 B.C., collected the mathematical knowledge of
that time in the 13-volume Elements. Problems are solved by
logical reasoning from an initial core of postulates (axioms),
a method commonly referred to as the classical axiomatic
method of Euclid. Alluding to the Elements, Euclidean
geometry is often referred to as elementary geometry.
Any theory of geometry that denies one or more of
the Euclidean axioms is a non-Euclidean geometry. Most
commonly a non-Euclidean geometry is a theory where the
Euclidean parallel axiom is rejected.
Congruence, Similarity, Correspondence
When superimposed, congruent figures can be made to
coincide exactly; thus, congruent figures have equal size and
shape, differing only in position.
= The notation = , or = , reads "congruent to".
Similar figures have the same shape, but differ in size; the
ratio of the lengths of any two corresponding lines in similar
figures is the same, and all corresponding angles are equal.
~ The notation ~ reads "similar to".
= The notation = reads "corresponds to". If 1 mm on a map
corresponds to a distance of 1 km, this can be written
1 mm = 1 km.
Planar
Plane Geometry
Solid Geometry
p. 381

377
11.4 Survey of Geometries
Euclidean geometry is based on definitions and axioms
described in Euclid's Elements. This geometry is mainly
concerned with points, lines, circles, polygons, polyhedra, and
the conic sections. The concepts of congruence and similarity
are fundamental in Euclidean geometry.
A geometry not based on the assumptions of Euclid is a
non-Euclidean geometry; in particular, a non-Euclidean
geometry does not depend on the parallel postulate of Euclid.
In learning the basics of mathematics, somewhat contrived
divisions of disciplines apply, while no such boundaries exist
for the mature user of mathematics.
p. 373 Projective geometry is the study of those properties of plane
figures that are unchanged when a given set of points is
projected onto a second plane.
p. 457 Trigonometry is the specialized geometry of the triangle,
originally defined in terms of the angles and sides of a right-
angled triangle but, as generalized, applicable to any type of
triangle.
Plane trigonometry is concerned with plane triangles,
spherical trigonometry with triangles on the surface of a sphere.
Trigonometric concepts are also widely used to simplify
calculations in fields of mathematics that are not primarily
geometrical.
p. 547 Analytic geometry investigates geometric problems by means
of coordinate systems, thereby transforming them into
algebraic problems.
Plane analytic geometry is devoted primarily to the analysis
of equations in two variables, solid analytic geometry to
equations in three variables. The methods are applicable to any
number of variables and to any dimensions.
p. 599 Vector analysis deals specifically with the study of quantities
that have both magnitude and direction. Geometrically, a
vector is drawn as an arrow in a specific direction, its length
representing the magnitude of the vector. This field includes
the study of flows and uses generalized methods of the
calculus. A generalization of vector analysis is tensor analysis, a
study of components, motions, and the like in n-dimensional
space.
Differential geometry applies differential and integral
calculus to curves, surfaces, and other geometrical entities.
p. 625 The mathematics of fractals is concerned with shapes having
generalized self-similarity, where patterns of the parts are
miniatures of the whole.
p. 378 Topology deals with those properties - often associated with
calculable invariant qualities - which are not altered by
continuous deformations.

378
Chapter 11 OVERTURE TO THE GEOMETRIES
11.5 Topology
Greek, homos, "same";
morphe, "shape", "form"
Mapping: p. 188
Topological Equivalence
Objects are topologically equivalent, or homeomorphic, if one
can be changed into another by topological transformation,
that is, by bending, stretching, twisting, or the like.
Objects X and Y are homeomorphic if and only if there is a
continuous one-to-one mapping from X onto Y whose inverse
mapping is also continuous. Informally we may think of
bending, stretching, or twisting; what is not allowed is folding
that brings once distant points into direct contact/overlap or
cutting unless followed by a regluing that reestablishes the
preexisting relationships of continuity.
.49
A sphere is topologically
equivalent to a cube or a handleless
mug but not to an anchor ring,
whereas an anchor ring is
topologically equivalent to a cup
or an iron; an object with two
holes is topologically equivalent
to a sugar bowl or a lidless teapot,
but not to a sphere or an anchor
ring.
Despite the distortion of the mirror, there is homeomorphism
between the dog and its reflection.

Section 11.5 Topology
379
Knots
Jordan Curve
JORDAN Camille
(1838 -1922)
VEBLEN Oswald
(1880-1960)
Genus: 0 1
Mathematically, knots are curves formed in space by first
interlacing a piece of string and then joining the ends
together. Knots may be classified according to the number of
crossings (overcrossing and undercrossing points). The
lowest possible number of crossings is 3. Knots of 3 and 4
crossings have only one type each, knots of 5 crossings have
two types, knots of 6 crossings have three types; thereafter the
number of knot types increases rapidly.
Two knots are topologically equivalent if, by continuous
movement (deformation) and not breaking the string, one
knot can be deformed into the other. In addition, just as a
triangle is equivalent (congruent) to its mirror image, a knot
and its mirror reflection are considered topologically
equivalent, although no continuous movement (deformation)
can change one of the two knots into the other.
The mathematical study of knots is an area of active research.
Practical applications are found in endeavors such as the
analysis of electrical circuits, molecular structure analysis,
and the planning of street and highway networks.
Genus of a Surface; the Jordan Curve
A continuous curve that does not intersect itself and has no
endpoints - e.g., a circle or a polygon - is a simple closed
curve, or Jordan curve; it divides the coordinate planes into
two regions, a theorem stated in 1893 by the French
mathematician Camille Jordan. The theorem is intuitively self-evident,
but a mathematical proof is another matter, and was not given
until 1905 by the American mathematician Oswald Veblen.
The genus of a surface is the largest number of Jordan curves
that can be drawn on the surface without cutting it into two
unconnected parts. For instance, the genus of a sphere is 0, for
any simple closed curve separates it into two parts. Similarly,
an ellipsoid and a convex polyhedron are of genus 0. An
anchor ring is of genus 1, a two-holed object of genus 2 - the
former always becoming disconnected by more than one cut,
the latter by more than two cuts.
A convenient means of determining whether a given point is
located inside or outside a convoluted Jordan curve is to draw a
straight line from that point to a point that is undoubtedly
located outside. If the line crosses the curve transversally an
odd number of times, the point is located inside the curve; if it
crosses an even number of times, it is outside. The crossing
number is used in computer graphics programs to decide
which pixels to shade when filling the inside of a region.
4

380
Chapter 11 OVERTURE TO THE GEOMETRIES
One-Sided Surfaces
The property of being two-sided or one-sided is a topological
invariant; thus, a one-sided surface cannot be changed into a
two-sided one by topological transformation, and vice versa.
A one sided surface may be of any genus > 1.
MOBIUS Augustus
(1790-1868)
The Mobius Strip
An interesting topological discovery, simple but surprising,
was made about 1865 by the German mathematician Augustus
Mobius, who showed that it is possible for a sheet to have only
one surface. This is
demonstrated by the Mobius
strip: Cut a long
rectangular strip of paper,
give the paper a half twist,
and paste the ends
together. The Mobius strip
has only one surface and
one edge.
The experiment can
continue: By cutting the
Mobius strip lengthwise in the middle, an edge is added and
we obtain a single twisted double-faced ring; thus, the Mobius
strip is of genus 1. A third lengthwise cut results in two
interlocking double-faced rings.
Fan belts and other belts for mechanical drives are sometimes
made with a half twist - as Mobius strips - to provide more
uniform wear.
KLEIN Felix
(1849-1925)
The Klein Bottle
Named after the German mathematician Felix Klein, the
Klein bottle has one surface with no edges; thus, the "bottle"
has neither an outside nor an inside.
© Trustees of the Science Museum, London
Topologically, cutting the bottle into mirror-symmetrical
halves forms two Mobius strips; vice versa, the edge of one
Mobius strip attached to the edge of its mirror-image twin
forms a Klein bottle. The Klein bottle is a one-sided suface of
genus 2.

381
11.6 Euclidean and Non-Euclidean Geometries
EVCLEIDIS
(c. 330-c. 275 B.C.)
Euclidean Postulate on Indefinite
Continuation of a Straight Line
Euclidean Postulates and Common Notions
Euclid's presentation of plane geometry is based on a number
of theorems that can all be derived from five postulates
(axioms) and five common notions (the phrasing below does
not always strictly follow Euclid's own).
The postulates are:
1. Exactly one straight line can be drawn between any two
points.
2. A straight line can be continued indefinitely.
Euclidean Parallel Postulate
eg**
EVCLIDIS
ELEMENTORVM
GEOMETRICORVM
LibriTredecim.
■ X TIADITIONI DOCTIIIIHI
N ASIRIDINI TVSINI
Nuncprimum Arabicc imprc&i.
lOMAE
I* TjfOgrMfbM MtJUiM .
KLD.XCIV.
3. With any point as center, a circle with any radius
may be described.
4. All right angles are equal.
5. Through a given point outside a given straight line,
there passes only one line parallel to the given line;
that is, such a line does not intersect the given line.
The common notions are:
1. Things equal to the same thing are equal.
2. If equals are added to equals, the wholes are equal.
3. If equals are subtracted from equals, the remainders
are equal.
4. Things which coincide with one another are equal.
5. The whole is greater than a part.
From his postulates and notions, Euclid deduced 465 theorems.
Non-Euclidean Geometries
For 2000 years, Euclid's system was held as the only possible
foundation of geometry, thought to be supported by logical
reasoning and empirical proof.
In the early 19th century, mathematicians demonstrated that
geometries that were just as valid and consistent as the age-old
Euclidean geometry could be produced by replacing two of
Euclid's postulates - the one about infinite continuation of a
straight line, and the one about parallel lines - by other
postulates that were employed with the remaining Euclidean
postulates and the common notions.

Chapter 11 OVERTURE TO THE GEOMETRIES
KLEIN Felix
(1849-1925)
The new concepts of geometry were later named hyperbolic
geometry and elliptic geometry. These names, founded on
analogies with conic sections, were suggested by the German
mathematician Felix Klein in his notable program for
classifying geometries (1872); for Euclidean geometry, Klein
coined the name parabolic geometry, a term which is, however,
rarely used.
LOBACHEVSKI
Nikolai Ivanovitch
(1793-1856)
BOLYAI Janos
(1802-1860)
GAUSS Carl Friedrich
(1777-1855)
POINCARE Jules Henri
(1854-1912)
Hyperbolic Geometry
Hyperbolic geometry substitutes the Euclidean parallel
postulate with the following: Through a given point outside
a given straight line pass more than one line not intersecting
the given line.
As one might expect, many theorems of hyperbolic geometry
contradict the theorems of Euclidean geometry. While in
Euclidean geometry the sum of angles of a plane triangle is
180°, in hyperbolic geometry it is less than 180° and varies
with the size of the triangle: the smaller the area of the
triangle, the closer the angle sum is to 180°. Two triangles are
similar in hyperbolic geometry only if they are congruent.
The first publication on hyperbolic geometry, by the Russian
mathematician Lobachevski, appeared in 1829. The
Hungarian mathematician Bolyai independently published
similar results in an appendix to a treatise written by his
father, with 1829 as the printing year, but apparently not
published until 1832. Lobachevski followed his first
publication with further writings on hyperbolic geometry, but
the works of Lobachevski and Bolyai were almost completely
neglected by their contemporaries.
Independently of Lobachevski and Bolyai, Gauss, the
prominent German mathematician, had already formulated a
hyperbolic geometry, but had decided that the results should not
be published in his lifetime. When they were published, about
thirty years after the writings of Lobachevski and Bolyai,
Gauss's fame made mathematicians take notice of the new
geometry and opened the door for appreciation of the works of
Lobachevski and Bolyai.
The French mathematician-physicist Jules Henri Poincare,
active in the entire field of mathematics, used the inside of
a circle as a model for hyperbolic geometry. In Poincare's
model, lines are either diameters or circular arcs whose ends
are perpendicular to the circumference of the circle; two such
lines which do not meet correspond to parallel lines.
If, in the circle to the left, P is assumed to be a point in the
hyperbolic plane and I a line that does not contain P, then an
infinite number of lines parallel to I can be drawn through P.

Section 11.6 Euclidean and Non-Euclidean Geometries
383
Elliptic Geometry
Elliptic geometry rejects the Euclidean parallel postulate on
the assumption that there are no parallel lines and, if extended
far enough, any two straight lines in a plane will meet.
In 1854, in a paper entitled Uber die Hypothesen, welche der
Geometrie zu Grunde liegen ("On the Hypotheses which Form
the Foundation of Geometry"), the idea of elliptic geometry was
RIEMANN Bemhard first conceived by the German mathematician Riemann; as a
(1826-1866) consequence, elliptic geometry is commonly referred to as
Riemann geometry, though this might also be taken to be the
more general geometry of Riemannian manifolds.
An important concept in elliptic geometry is that
boundlessness does not necessarily imply infinitely long lines: If we
follow a straight line we do not come to an end, nor would we if
we followed the equator of the Earth. Thus, experience does not
convince us of the infinitude of a straight line - it only tells us
that a straight line is boundless. This concept, that space could
EINSTEIN Albert be unbounded without being infinite, was adopted by Albert
(1879-1955) Einstein in his general theory of relativity (1916).
In Riemann geometry, all perpendiculars to a straight line
meet at a point, and (as in hyperbolic geometry) triangles that
are similar are also congruent.
Whereas in hyperbolic geometry the sum of the angles of any
triangle is less than 180°, in Riemann geometry the sum of the
angles of any triangle is greater than 180 °.
Elliptic Geometrist:
- If 'Euclid ivere ative today, do you thinf^ he zuould be fooled
upon as a remarkable person?
Hyperbolic Geometrist:
- *By all means. 9ie would be over 2000 years old.

384 Chapter 11 OVERTURE TO THE GEOMETRIES
The Infinitude of Geometries
There is no such thing as only one unassailable,
mathematically true, geometry. From a mathematical viewpoint, any
geometry - or any other branch of mathematics - that does not
produce contradictions is acceptable. Another matter of
concern is, however, to find the geometry that gives the most
accurate representation of the physical world.
Despite its inherent flaws, Euclidean geometry is still the
basis for most practical applications of geometry - it has taken
human beings to the Moon and beyond.

385
Chapter
12
ELEMENTARY GEOMETRY
Page
12.0 What Do We Mean by "Elementary Geometry"? 386
12.1 Geometric Elements and Figures 386
12.2 Units of Measurement 409
12.3 Euclidean Construction 413
12.4 Theorems and Formulas 425
12.41 Plane Geometry 426
12.42 Solid Geometry 446

386
Chapter 12 ELEMENTARY GEOMETRY
12.0 What Do We Mean by "Elementary Geometry"?
p. 381
The main topic of this chapter is Euclidean geometry. Based
on definitions and axioms described in Euclid's Elements,
Euclidean geometry is sometimes referred to as elementary
geometry. This chapter is also elementary, or basic, in the
sense that the mathematical function concept is not required,
and a coordinate system is not used.
12.1 Geometric Elements and Figures
We will study concepts and names of geometric elements and
figures, a sometimes dreary pursuit but giving the necessary
preparation for more gratifying study, such as geometric
construction (Section 12.3) and theorems (Section 12.4).
Points
X
o
Cusp
Node, Crunode
Salient point
A geometric point has no dimension - is void of quantity -
and therefore cannot be drawn as "just a point". Thus, the
concept of a geometric point is axiomatic.
A point is often represented by a small cross - signifying the
intersection of two lines - or a small circlet.
Points associated with plane curves often have special names;
see, e.g., the definition of tangent, on page 388.
A cusp is a double point on a curve where the curve has two
coincident tangents.
A cusp of the first kind is a point where the curve has a branch
on each side of the common tangent near the point of contact.
A cusp of the second kind is a point where the two parts of the
curve lie on the same side of the common tangent near the
point of contact.
A double cusp - or point of osculation - is a point where the
two branches of the curve have a common tangent, each branch
extending in both directions of the tangent.
A node, or crunode, is a point where two branches of a curve
cross and have different tangents.
At a salient point two branches of the curve meet and stop, and
have different tangents.

Section 12.1 Geometric Elements and Figures
387
Locus
A locus (plural: loci) is a geometric figure for which all of its
points satisfy a given condition. For instance, the locus of all
points in a plane at equal distance from a given point is a
circle; the locus of all points in space with the same distance
from a given point is a sphere.
Lines
Dual Elements
Dual Operations
-*-
-*■
<U
Transversal
Perpendicular
Latin, per, "through
pendere, "to weigh"
Crossing Line
Skew Line
Ray (Half-Line)
Origin
*
Q
Curve, Arc
c?-
A moving point describes a line that has only length, but no
breadth. As with the point, the concept of a line is axiomatic.
We find ourselves with the slightly paradoxical truth that
two nonparallel lines define a point
and
two points define a line;
in this sense points and lines are said to be dual elements, and
the intersection of lines to give points and the connection of
points to give lines are dual operations.
A straight line - usually called just a line - is the shortest
distance between two points. It may extend without limit, or be
a determinate straight line - also called a line segment -
which contains both of the endpoints and all points between
them; the length of the line segment is the distance between the
endpoints.
A succession of straight line segments is a broken line.
Two or more lines in the same plane are parallel lines if they
do not meet however far they extend.
A line that intersects two or more (often parallel) lines is a
transversal.
A line that meets another line at right angles is a
perpendicular, indicated by a small square in the angle.
Lines which lie in different planes and do not intersect each
other are crossing lines or skew lines.
A straight line issuing from an initial point, often denoted O
for origin, is a ray or a half-line; a closed ray includes the
origin, an open ray does not.
A curve is any continuous image of a line or segment; an arc
is a portion of a curve, as in an arc of a circle. Curves are
usually curved, but they can have straight sections.
A straight line that intersects a curve is called a secant; the
part of the secant contained between the points of intersection is
a chord.
tangent

388
Chapter 12 ELEMENTARY GEOMETRY
Tangent
Inflection Tangent
Perpendicular
Normal
A straight line that just touches a curve - a limit position of the
secant - is a tangent; it has a double point of contact with the
curve.
point of inflection
secant
tangent
A tangent to a curve in a point of inflection - where the curve
changes its direction of curvature - is an inflection tangent;
it has three points in common with the curve (these notions are
best understood in the context of calculus, which will be
discussed in later chapters).
A straight line at right angles to the tangent in its point of
contact with the curve is a perpendicular to the tangent and a
normal to the curve.
Angles
z
Latin, vortex, "whirl", "summit",
"top of the head"
null
angle
zero
angle
>
0°
Orad
acute
angle
right
angle
P-
A plane angle (Z), or simply an angle, is formed by two rays
- sides or legs of the angle - which extend from a common
point, the vertex of the angle.
Angles have different names:
90°
7c/2 rad
obtuse
angle
straight
angle
flat
angle
reflex angle
<?
270°
3 k/2 rad
full angle
or
full circle
©■
360°
2 n rad
Two angles with a leg in common are adjacent angles.
Complementary Angles
Supplementary Angles
Two angles whose sum is a right angle are complementary
angles; two angles whose sum is a straight angle are
supplementary angles:

Section 12.1 Geometric Elements and Figures
389
Bisector
Vertical Angles
A bisector is a straight line which divides an angle into two
equal angles.
Two intersecting straight lines form two pairs of vertical
angles; vertical angles are equal.
vertical angles ,
a - a
P = P>
These angles are called vertical because each side of one is an
extension through the vertex of a side of the other.
A transversal which intersects a pair of parallel lines
produces many pairs of angles, some of which are equal, as
shown below:
are exterior angles
are interior angles
are alternate interior angles
are alternate exterior angles
«1
«2
n
«i
«i
n
«i
«2
Pi
p2
= «2;
= 72;
= «2;
= 72;
= 71;
= 72;
72
71
5i
Pi
Pi
Si
Pi
P2
82
Si
= &
= ¾
= /fel
= Si\
= <5il
= s2\
are corresponding angles
are opposite angles
* * *
Joe was out hoeing beans when
he noticed smof^e from the hay
field. He grabbed two budgets
from the shed and, via the edge
of the irrigation ditch, toof^ the
shortest way to the burning
hay. Living and worlqng by the
ma^im "The shortest distance
between two points is a straight
Cine", how did Joe set his
course?
[Answer on page 391]

Chapter 12 ELEMENTARY GEOMETRY
Ellipses and Circles
Ellipses
An ellipse is a plane curve where the sum of the distances
between two fixed points (the foci) and any point on the
periphery is constant; below, i*\ and F2 are fixed points and Px
and P2 are arbitrary points on the periphery:
Ql+Ql" = Q2+Q2"
Major Axis
Minor Axis
An ellipse has two axes of symmetry, a major axis, between the
vertices of the ellipse, and a minor axis, which intersect at the
center of the ellipse:
vertex
vertex
Perimeter
Periphery
Circumference
The boundary of a figure in a plane is the perimeter. A curved
perimeter, such as is found in an ellipse, is called the
periphery.
The circumference is the length of a perimeter or periphery.
Circles
Arc
Periphery
A circle is a plane curve that is the locus of all points in the
plane equidistant from a given point, the center of the circle.
An arc of the circle is bounded by two distinct points on the
periphery.

Section 12.1 Geometric Elements and Figures
391
Secant
Chord
Diameter
Tangent
Radius
Sector
Segment
Subtend: to be opposite to
and mark off
A secant of a circle is a straight line intersecting the periphery
in two points; a chord is the part of the secant within the circle.
A chord passing through the center is the diameter- the longest
chord of the circle.
A tangent is a straight line which has a double point of contact
with the periphery, and may be regarded as the limit position of
a secant. The tangent is perpendicular to the radius of the
circle at the point of contact.
A sector of the circle is bounded by two radii and the included
arc; a segment is bounded by a chord and the arc subtending
the chord:
Angle Subtended by an Arc
A central angle, 0, subtended by an arc, a, as shown below, is
an angle whose vertex is at the center and whose sides are
radii. The peripheral angle, y/, subtended by the same arc a,
has half the measure of 6:
Angle Subtended by a Chord
The angle a subtended by a chord b has its vertex on the
periphery and its two sides are chords that together with b form
a triangle as shown below. Because a is the peripheral angle
for the arc /3, its measure is independent of where its vertex
falls in the complement of /3.
* * *
Answer to the haystack
problem from page 389: To hit the
point on the edge of the
irrigation ditch that gives the
shortest total distance to the
haystack, Joe envisioned a
mirror image of the
haystack (the edge of the ditch
closest to Joe being the
"mirror") and a straight line
between the shed and the
mirror image.
<*,-'

392
Chapter 12 ELEMENTARY GEOMETRY
Polygons
Greek, poly, "many"; A polygon is a plane figure with three or more angles and as
gonia, "angle", from gdny, "knee" many sides. It is bounded by a broken line forming a simple
closed curve, a circuit without self-intersections.
Diagonal
A diagonal is a straight line connecting two opposite vertices.
diagonals
Polygons are named after the number of sides or vertices:
No. of
sides
3
4
5
6
7
8
Name
Triangle
Quadrilateral
Tetragon
Pentagon
Hexagon
Heptagon
Octagon
No.
side
9
10
11
12
n
of
JS
Name
Nonagon
Decagon
Undecagon
Dodecagon
n-cron
Regular Polygon
Convex
Concave
Reentrant Angle
Salient Angle
In a regular polygon, all sides have the same length, and all
interior angles are equal.
In a convex polygon each interior angle is less than a flat
angle (180°); a polygon is concave if an interior angle
exceeds 180°:
Convex Hexagon
Concave Hexagon
The inward-pointing angle of the concave polygon is a
reentrant angle; the other angles are salient angles.
Triangles
Greek, skalinos, "uneven"
Greek, iso-, "equal", skelos, "leg"
A triangle is a polygon with three sides. Every triangle is
contained in a plane. If no two sides are equal, it is a scalene
triangle; if two sides are equal, it is an isosceles triangle;
and if all sides are equal, it is an equilateral triangle.
a
Scalene Triangle
Isosceles Triangle Equilateral Triangle

Section 12.1 Geometric Elements and Figures
393
If each interior angle of a triangle is less than a right angle,
we speak of an acute triangle; if one angle is greater than a
right angle, we have an obtuse triangle. Acute and obtuse
triangles are called oblique triangles. A triangle with a right
angle is called a right-angled triangle or a right triangle.
Acute Triangle
Right Triangle
Obtuse Triangle
Egyptian Triangle
Leg Base
Hypotenuse
Greek, hypoteinoysa,
"stretching under"
Median
A right triangle with the sides of 3, 4, and 5 units is known as
an Egyptian triangle.
The two equal sides of an isosceles triangle are known as the
legs, the third side being the base. In a right triangle, the side
opposite the right angle is the hypotenuse, the other two sides
being the legs.
hypotenuse
A median of a triangle is a straight line from a vertex to the
midpoint of the opposite side:
a
Height, Altitude
Normal, Perpendicular
The height, or altitude, of a triangle is a normal, or
perpendicular, from a vertex to the opposite side or to its extension. In
the triangle ABC below, ha is the height from vertex A to the
extension of side a; h^ is the height from vertex B to side 6; and
hc is the height from vertex C to the extension of side c:

394
Chapter 12 ELEMENTARY GEOMETRY
Quadrilaterals
Quadrangle
Tetragon
Trapezoid, Trapezium
A quadrilateral is a polygon with four sides; other names, less
common, are quadrangle and tetragon.
A quadrilateral with no two sides parallel is known as a
trapezoid in the U.K., a trapezium (plural: trapezia) in the
U.S.; if two sides are parallel, it is a trapezium in the U.K.
and a trapezoid in the U.S.
base
Leg
Isosceles Trapezoid
The base of a trapezium or trapezoid is generally its longest
side, on which it is supposed to stand. If two sides are parallel,
they are usually both called bases.
The height, or altitude, of a trapezium or trapezoid is the
perpendicular from the highest point toward the base, or the
distance between the two parallel bases.
The two nonparallel sides are called legs. If the legs are of
equal length, we have an isosceles trapezium or trapezoid:
median-
Median
A median, or midline, is a straight line segment joining the
midpoints of the nonparallel sides.
Rhomboid
Parallelograms
A general parallelogram is a quadrilateral in which both
pairs of opposite sides are parallel.
A rhomboid is a parallelogram whose adjacent sides are
unequal:
Rhombus
A rhomboid with all sides equal is a rhombus (plural: rhombi)

Section 12.1 Geometric Elements and Figures
395
Rectangle
Square
A rectangle is a right-angled parallelogram; a square is a
rectangle with all sides equal - a regular quadrilateral:
a
a
a
a
a
a
* * *
How zuouCd you construct a quadriCaterat that, after joining the,
midpoints of its adjacent sides, includes a paraCCeCogram?
Answer: jfpsmofi sofyooj puv'fuj, -op ))m pmvjuptmb Ivuy
Kites and Deltoids
Kites and deltoids are quadrilaterals whose adjacent sides are
equal in pairs.
kite
deltoid
Kites and deltoids have mirror symmetry, as do rhombi.
--^ ^
W \\
m
/A/ A A 11,1 \ J\i,
Tiling
The mathematical study of tiling (tessellation) is concerned
with how shapes can be placed to completely fill a plane or
space.
The only regular polygons that - one kind at a time - can tile
the plane are equilateral triangles, squares, and hexagons
(cf. honeycombs and the crystal pattern of snowflakes, both
originally investigated by Kepler).
By allowing two or more types of regular polygons to meet at
each vertex but requiring the vertex configurations to be the
same, you allow an additional nine tilings; two of these are
mirror images of each other. Allowing more general shapes
leads to a wealth of possibilities.
Tiling - in the plane and in space - forms a basis of active
and challenging mathematical research with applications in
graphics, networks, the study of chemical structure, and
geology.

396
Chapter 12 ELEMENTARY GEOMETRY
Three-Dimensional Figures
Polyhedra
STOR Lorenz
(? - c. 1621) p. 373
Greek, poly, "many";
hedra, "base"
Convex Polyhedron
Concave Polyhedron
Edge
Vertex
Face Diagonal
Space Diagonal
>CTA!OROH EtEAo/ATYtt SOUUVM
Title page of Lorenz Stor's Geometria et Perspectiva (1567).
A solid whose faces are plane polygons is known as a
polyhedron (plural: polyhedrons, polyhedra). Polyhedra are
named according to the number of faces: a polyhedron with
four faces is a tetrahedron; one with six faces is a hexahedron;
and so on.
A convex polyhedron lies entirely on one side of a plane that
contains any one of its faces; a concave polyhedron has at
least one face so located that there are parts of the polyhedron
on both sides of a plane containing that face.
The faces of a polyhedron intersect each other along the edges,
which meet at the vertices, or corners. A straight line joining
two vertices of a face is a face diagonal; a line between two
vertices that are not in the same face is a space diagonal.
face
diagonal
space
diagonal

Section 12.1 Geometric Elements and Figures
397
Regular Polyhedra
Platonic Solids
PLATON
(427-348 or 347 B.C.)
Regular Polyhedra
The faces of a regular polyhedron are identical regular
polygons, each of whose vertices are surrounded by congruent
arrangements of faces. Taken together, the faces intersect
only along their edges and surround a topological ball.
There are five regular polyhedra, all of them convex, often
referred to as Platonic solids in a tribute to Plato, who used
them as metaphors in his cosmology: the tetrahedron, the
hexahedron or cube, the octahedron, the dodecahedron, and the
icosahedron.
Hexahedron (Cube)
Octahedron
Dodecahedron
Icosahedron
Tiling: p. 395
The cube is the only regular polyhedron to completely fill
space.
The Pythagorean brotherhood, about 500 B.C., was originally
aware of only four regular polyhedra - tetrahedron, cube,
octahedron, and icosahedron - considered to represent the four
basic elements fire, earth, air, and water. Measuring vessels
from the Egyptian Old Kingdom (3110-2884 B .C.) provide
evidence, however, that all five regular polyhedra had been
known long before the Pythagorean brotherhood.
fire
air
earth
moist
water

398
Chapter 12 ELEMENTARY GEOMETRY
When the Pythagoreans realized that there was also a fifth
regular polyhedron, the dodecahedron, they considered it
to represent a fifth element (quinta essentia) - aether or
universe.
Quinta essentia
dodecahedron
KEPLER Johannes
(1571 -1630)
Pythagorean and Platonic ideas continued to rule for
centuries - the universe and its components had to be in
beautiful and harmonious order, as in the geometric forms of
circles and Platonic solids. A notable example is Johannes
Kepler's Mysterium cosmographicum, published in 1596, in
which Kepler described our solar system as having the Sun in
the center of six circular planetary orbits (only six of the nine
planets were known at that time); the Platonic bodies, one
within the other, could be fitted between the orbits of the planets.
ASTRONOMIA NOVA
AITIOAOTHTOS,
S E V
PHYSIC A COELESTIS,
tradica commcnciriis
DE MOTIBVS STELLiE
M A R T I S,
Ex oblervacionibiu C.
TTCHONIS BRAHE;
Juflu 6c fumpcibus
RVDOLPHI II.
ROMANORVM
IMP ERATO MS ice:
TABVtAfIioR.BIVNfPLANtTAlt.VM DIMENSIONS. XX DttTANTIAS FULqVINqVE
ILLVSTRISS: PR.INCIPI.AC DNO DNO FklDERlGO. DVCI WIRl
rtNtM&KO.ET TlitCiO. C°MIT! MONTI* KCLOAB.VM E1C.C0NSCCILATA
Plurium annonun pertimci ftudio
i annoram pertiiu
clabonu rragz ,
•4,5. C . Of> f». dXithtautle*
JOANNE KEPLEHO,
Axmo nx Dioay(i»nx cU la e ix.
From Johannes Kepler's
Mysterium cosmographicum (Tubingen, 1596).
In his Astronomia nova of 1609, Kepler demonstrated that the
planet Mars travels in an elliptic orbit with the Sun in one of
its two foci. He stated that, like the Earth, the "heavenly"
planets were material bodies and, thus, did not have to revolve
along "perfectly" circular paths. Instead of the previously
assumed equal lengths of arcs of a circle for equal intervals of
time, Kepler found an alternative form of harmony: with the
Sun in one focus of the elliptic orbit, the planet would sweep, in
equal intervals of time, equal areas of the ellipse.

Section 12.1 Geometric Elements and Figures
399
Leonardo da Vinci's
illustration of a semi-
regular polyhedron
in Luga Pacioli's De
Divina Proportione
(1509).
See text above, right.
LEONARDO DA VINCI
(Italien painter,
sculptor, anatomist, architect,
engineer; 1452 - 1519)
Semi-Regular Polyhedra
Like regular polyhedra, semi-regular polyhedra are bounded
by regular polygons, but of more than one kind, generally two
kinds, and they can all be inscribed in a sphere.
The polyhedron formed by 12 pentagons and 20 hexagons - a
modern soccer ball - is illustrated to the left; it was described
by Archimedes among a total of 13 semi-regular polyhedra,
illustrated below in a reproduction from the Harmonices
mundi (1619) by Johannes Kepler.
it
v.
I
oannis
Kcpplcrl
HARMONICES
M V N D I
LIBRI V. Qvoivm
Primus G comitmcvs, DeFigurarumReeuIarium-, quit Proporcio*
ne»Harmofucaicon/Uruunr»ortuic demonlirarionibus.
ScCUnJuSA>XHtTtCTOMlCrS,/eUCxGtCMtTB.IA FlOVLATAtDeft*
gurarum Regularium Congrucntia in piano vel Cclxdo:
Tcrcius propneHA».MOKicr* De Proporcionum Harmonica rumor-
cucxfiguris»dcque Narura fie Differ en riisrerum adcanrum per*
anenaum, coacrc Vetera:
Quirt us MiTAfKYucri, Pstchologicts fie AmoLocicrs, De Har-
monurummcaali Etfcnaa carumquc gcncribus*n Mundoi pnr/er-
omdc Harmooia radionim, ex corponbas«r!c/bbus in Tcrramde.
Jcendenubas, tiu/que effeda in Nauira feu Anima fublunari fie
Humana:
Qoincua Aitiokomictj ic Mxtaf htjioj • De HinnohiU ab(bluai&*
mis mocuum ctrlcftiam, orraque Ecccnaiciarum ex proportion!*
bos Harmonicu.
Appendix tuber, compararioncm hums Opens cam Harmonica Q.
Pcolcmcs Lbro 11 Lcumque Roberade Flu&ibtu.didi Flud-Medici
Oxonicnus fpcculaaonibus Ha/moaicui opcri de Macroco/mo fie
Mkroco/mo m/crcis.
CmmS.C.W.PnmUti*U4*tus XV.
Lincii Au ftria:,
SumpubrnGoDoniD! TAMPACHiiBibLFiancoC
ExcudcbacIoANNis Plancvs.
•4 HH o &t. TC. XIX.
/\ \
/ 1 \ 1 \
/ 1 \ 1 \
/ 1 _ \ 1 \
/ 1
L /
\ \ \ \
\ ) \ d
\ 1 \ / \ /
Y r r
\ A A
i \ / \ / \
\ / y- -/— ■
\ \
\
\
\
\
\
/ t
/
/
i
i
1
/ /
/
/
/
/
/
HAUY Rene-Just
pronounced "Ah oui")
(1743 -1822)
Another type of polyhedron
bounded by equal polygons
occurs in crystallography.
An example is the rhombic
dodecahedron, described in
1822 by the French crystal-
lographer Rene-Just Haiiy.
The cube is the nucleus of
the rhombic dodecahedron.
From Hauy's Traite de
christallographie (1822).

400
Chapter 12 ELEMENTARY GEOMETRY
Prisms
Base
Lateral Face
A prism is a polyhedron with two congruent and parallel faces
(bases), whose remaining faces (lateral faces) are
parallelograms:
Parallelepiped
Rectangular Parallelepiped
Cube
Right Prism
Oblique Prism
Regular Prism
Altitude, Height
Truncated Prism
Altitude, Height
Frustum
Latin, frustum, "a piece"
K-:-,«.>MW«.:.w.y.:.v.v.v.v.',
I
I
I
wmmmm
■•■•■•■•|......:::>::....y.y.%w
I
I
I
I
I
Jm
'.•.•.•■•■y.v.'l
ifitifffii
Prisms are named after the shape of their base faces: a prism
whose base is a triangle - for instance, a wedge - is a
triangular prism; a prism with a pentagon as a base is a
pentagonal prism; and so on. A parallelepiped is a prism with
six faces, all parallelograms. A rectangular parallelepiped is
bounded by six rectangles, a cube by six equal squares.
The lateral faces of a right prism are perpendicular to the base;
those of an oblique prism are not. A regular prism is a right
prism whose bases are regular polygons.
The altitude, or height, of a prism is the perpendicular distance
between the bases.
A truncated prism is the portion of a prism contained between
the base and a plane that is not parallel to the base:
Pyramids
A pyramid is a polyhedron whose base is a polygon and whose
lateral faces are triangles with a common vertex.
A pyramid may be triangular, quadrangular, pentagonal,
etc., depending on the shape of its base. The altitude, or height,
of a pyramid is the perpendicular distance from vertex to base.
A regular pyramid has a regular polygon for its base; the
height meets the base at its center. A frustum (plural: frusta)
of a pyramid is the part between the base and any transecting
plane parallel to the base.

Section 12.1 Geometric Elements and Figures
401
Cylinders
Base
Lateral Surface
Generator
Generatrix
(feminine form of generator)
Directrix
(feminine form of director)
A cylinder is a solid with three faces. The two bases are
parallel plane surfaces having the shape of any closed plane
curve. The third face, or lateral surface, is generated by a
moving straight line, the generator or generatrix, which
traces the perimeter of the base, the directrix, at the same time
keeping parallel to its original direction in space.
directrix
generatrix
Right Cylinder
Oblique Cylinder
The lateral surface of a right cylinder is orthogonal to the
bases; other cylinders are oblique cylinders. The height, or
altitude, of the cylinder is the perpendicular distance between
the bases.
• v.v.w.v.y.vN
Right Circular Cylinder
Oblique Circular Cylinder
Cylinders are also classified according to the shape of their
bases: circular, elliptic, etc. Cylinders may be hollow, having
one or more holes parallel to the generatrix; a cylinder with
one coaxial hole is a pipe, or a tube.
W.l.w..lW.I.'AMi'.'ii.!l!W
kUiUtiliiilSlUili
Right Elliptic Cylinder

402
Chapters ELEMENTARY GEOMETRY
Cones
Generator, Generatrix
Vertex, Apex
Directrix
Nappe
French, nappe, "sheet", "surface"
Half-Cone
Base
Lateral Surface
Axis
Height
Right Cone
Oblique Cone
Right Circular
Cone
Frustum
Latin, frustum, "a piece"
A conical surface is generated by a straight line (the generator
or generatrix) passing through a fixed point (vertex, or apex)
which moves so as to trace a given curve (the directrix). The
two parts of the conical surface are called nappes (singular:
nappe) - one on either side of the vertex.
generatrix
The nappe is also known as a half-cone; when there is no risk
of ambiguity, it is generally called just a cone.
A cone is thus bounded by a plane surface, the base, which may
be any closed curve, the directrix, and a nappe which forms the
lateral surface of the cone.
A line from the vertex of the cone to the center of its base is the
axis of the cone. The height, or altitude, is the perpendicular
distance from vertex to base (see illustrations below).
In a right cone the base is perpendicular to the axis; in an
oblique cone it is not. Like cylinders, cones are classified
according to their bases - circular, elliptic, etc.
Oblique Circular
Cone
Right Elliptic
Cone
Frustum of a
Cone
The frustum (plural: frusta) of a cone is a part between the
base and a plane parallel to the base.

Section 12.1 Geometric Elements and Figures
403
Solids of Revolution
Axis of Rotation
pp. 401, 402
A plane figure revolving around a straight line in the plane
- the axis of rotation - generates a solid of revolution.
A right circular cylinder and a right circular cone, already
discussed, are examples of solids of revolution.
Radius
Secant
Chord
Diameter
Radius
Great Circle
Small Circle
Tangent
Spherical Segment
Spherical Cap
Diametral plane
Hemisphere
Spherical Layer
Spherical Zone
Altitude, Height,
Sphere
A sphere is a solid generated by a circle that revolves about its
diameter. It has only one surface; all points of the surface are
at the same distance - the radius - from the center.
chord
A secant is a straight line that cuts the surface of the sphere in
two points; the portion of the secant that lies between these
points is a chord. The longest chord, passing through the
center of the sphere, is the diameter. The distance from the
center to the spherical surface is the radius of the sphere.
If a plane transects a sphere through the center, the intersection
will be a great circle; an off-center intersection constitutes a
small circle.
A tangent to the sphere is the tangent of the periphery of a great
circle.
altitude
A plane that cuts a sphere divides it into two spherical
segments and its surface into two spherical caps. If the plane
is a diametral plane, cutting along a great circle, the segments
are hemispheres.
Parallel planes divide the sphere into several segments; a
segment between two parallel planes is a spherical layer. The
spherical part of its surface is a spherical zone.
The thickness of a spherical segment, or layer, measured at
right angles to the transecting planes, is its altitude, or height.

Chapter 12 ELEMENTARY GEOMETRY
cap
sector
wedge
digon
lune
Spherical Wedge
Lune, Digon
Spherical Polygon
Semilune
Two diametral planes that form an angle divide the sphere
into two spherical wedges, opposite wedges being
congruent. The surfaces of wedges are spherical lunes, or digons
(cf. German Zweieck, "two-corner").
A radius that moves along a circle on the surface of a sphere
generates a spherical sector, consisting of a spherical
segment and a right circular cone. If the contour is a great circle,
the sectors become hemispheres.
A part of a spherical surface bounded by arcs of three or more
great circles is a spherical polygon; a special form is the
spherical triangle - and the semilune is a special form of
spherical triangle.
semilune
semilune
Spherical Excess
The sum of the angles of a spherical triangle is always greater
than 180°, and may be as high as 540°; a spherical triangle
may have two or even three right angles or obtuse angles. A
spherical triangle with two right angles is called birectangu-
lar, one with three is trirectangular.
Three mutually perpendicular planes passing through the
center of a sphere divide the surface of the sphere into eight
trirectangular spherical triangles.
Spherical excess is the sum of the angles of a spherical
triangle, less 180°.

Section 12.1 Geometric Elements and Figures
405
Ellipsoid
Prolate Ellipsoid
Oblate Ellipsoid, Spheroid
;***#•»■
.*<*&.
\-&.T-
An ellipse rotating about
its major axis generates
a prolate ellipsoid of
revolution; if rotated
around its minor axis,
an oblate ellipsoid of
revolution will develop.
Another term for oblate
ellipsoid of revolution is
spheroid.
A "geoid" would emerge if the Earth were altered by removing
all parts above mean sea level and filling in all gaps below
mean sea level. As the Earth is "flattened at the poles", the
geoid fits the description of a spheroid.
Paraboloid
A paraboloid of
revolution is formed by
revolving a parabola about its
axis.
Torus
meridian
Meridian
Parallel
A torus (anchor ring or
doughnut) is generated
by a circle rotating on
an axis outside it but in
the same plane.
A plane containing the
axis of rotation
intersects the surface of the
torus, or any other solid
of revolution, along
meridians; a plane at right angles to the axis intersects the
surface along parallel circles, or parallels for short.
parallel
circles

406
Chapter 12 ELEMENTARY GEOMETRY
Conic Sections
Circle
Focus
Directrix
Eccentricity
Ellipse
Parabola
Hyperbola
This family of curves may be described by slicing a cone as
shown above, or as
the locus of all points in the plane whose distances
- r from a fixed point, the focus; and a from a given
straight line, the directrix - have a constant ratio.
This ratio is called the eccentricity and is denoted e:
r
— = e .
a
We distinguish four cases, depending upon the numerical
value of e:
e = 0 corresponds to a circle
0 < e < 1 corresponds to an ellipse
e = 1 corresponds to a parabola
e > 1 corresponds to a hyperbola
circle
ellipse
parabola
hyperbola
-5--°
a
o<£<i
^=1
■5->1
The relation r la = 0 for the circle shows that the directrix is
placed at infinite distance a = <>° from the circle.

Section 12.1 Geometric Elements and Figures
407
The ellipse and hyperbola have an extra focus and directrix;
for reasons of symmetry, the definition will apply also for
them.
Strictly speaking, the parabola also has two foci and two
directrices, one pair of which has receded into infinity.
The eccentricity of the circle is zero, e = 0. If the eccentricity is
allowed to increase from near zero - corresponding to an
ellipse closely resembling a circle in shape - the ellipse will
continually lengthen until the right-hand side of it vanishes
into infinity, and the ellipse turns into a parabola, e = 1, with
just one open branch.
e = 0
e = l/2
e = 3/4
e = l
When the eccentricity increases still further, the "lost" right-
hand end of the ellipse makes a reappearance "from the other
side of infinity", as it were, and turns into the left-hand
branch of a hyperbola.
e = 3/4
e = l
= 5/4
We can formulate another set of definitions for the curves,
spelled out individually:
The curve of
- the circle
- the ellipse
- the parabola
- the hyperbola
is the locus of
- all points that have the same distance
from a given point, the center of the
circle;
- all points whose distances to two
given points, the foci, have a constant
sum;
- all points that have the same distance
to a given point, the focus, and a given
line, the directrix;
- all points whose distances to two
given points, the foci, have a constant
difference.
The definitions given for the ellipse and the hyperbola apply
also to the circle and the parabola, if we understand that the two
foci of the circle coincide and that the second focus of the
parabola is located at infinite distance:
oo
±r =
oo

408
Chapter 12 ELEMENTARY GEOMETRY
We come full circle to again view those curves as conic
sections - or conies - produced by a plane slicing a circular
cone at different angles to the axis of that cone.
A plane cutting the cone
at an angle
- orthogonal to the axis =>
- between right angles to the axis
and parallel to the generatrix =>
- parallel to the generatrix =>
- between a plane parallel to the
generatrix and a plane parallel
to the axis =>
will produce
a circle
an ellipse
a parabola
a hyperbola
Circle
Ellipse
Parabola —'
Hyperbola
APOLLONIOS
(c. 255 -170 B.C.)
The name conic sections derives from the 8-volume treatise
Conies (KcoviKdx) by Apollonius of Perga, renowned
mathematician and astronomer of antiquity. Of the work, 7 volumes
are still extant (Books I - IV in the original Greek, Books
V - VII in Arabic translations); some ideas of book VIII may
be gained from the encyclopedic work of Pappus (4th century).
Apollonius is also the originator of the names ellipse,
parabola, and hyperbola.
The curves have another geometric property that can be made
use of for the reflection of light rays and beams of sound.
The curve of
- the circle
- the ellipse
- the parabola
- the hyperbola
reflects rays issuing from a/the focus
- back to the center of the circle;
- into the other focus;
- as a parallel outgoing beam;
- as if coming from the other focus.
circle
ellipse
parabola
hyperbola

409
12.2 Units of Measurement
Length, Area, and Volume
Length
meter In the International System of Units (SI), the
basic unit of length (I) is the meter (m), defined
("ISO 31-1:1992"):
"The meter is the length of the path travelled by
light in vacuum during a time interval of
1/299 792 458 of a second."
1 foot = 12 inches = 0.305 m
lyard = 3 feet = 0.914 m
1 mile (statute) = 5 280 feet = 1.609 km
1 mile (nautical) = 1.852 km (= 1 minute angle on a great
circle)
Area
square The derived unit of area (A) is the square meter
meter (m2):
"1 square meter is the area of a square with sides
of length 1 meter."
1 acre = 43 560 square feet = 4 047 m2
1 square mile = 640 acres = 2.590 km2
Volume
cubic The derived unit of volume (V) is the cubic meter
meter (m3):
"1 cubic meter is the volume of a cube with edges of
length 1 meter."
1000 liters (1) = 1 m3
1000 milliliters (ml) = 11
1 fluid ounce (British Imperial) = 28.413 ml
1 fluid ounce (U.S.) = 29.573 ml
1 pint (liquid and dry, British Imperial) = 568.245 ml
1 pint (liquid, U.S.) = 0.473 1
1 pint (dry, U.S.) = 0.5511
1 quart = 2 pints
1 gallon (British Imperial, liquid or dry; U.S., liquid
measurement only) = 4 quarts, corresponding to
British Imperial: 4.546 1
U.S.: 3.7851

410
Chapter 12 ELEMENTARY GEOMETRY
Angles
Plane Angles
There are several systems of units currently in use for
measuring and expressing plane angles, all of them derived from
the division of the full circle.
Radian
The radian is the standard angular measure in
the International System of Units (SI):
"1 rad is the angle between two radii of a circle
which cut off on the circumference an arc equal
in length to the radius."
s
s
Degree
"minutes" = small units
"seconds" = "second-order minutes"
or "second minutes" or, briefly, "seconds"
Gon
A full angle is 2n radians.
The degree is a non-SI unit approved in 1969 by the
General Conference on Weights and Measures.
An established unit with a history of over 4 000
years, dating from the ancient Babylonian
astronomers, the degree is l/360th of the full
circle. It is divided into 60 minutes, each of 60
seconds:
1° = 60'; 1' = 60"
For ease of calculation, the degree may also
be divided decimally, the degree symbol either
taking the place of the decimal marker (e.g.,
112°37, or being placed after the last decimal
digit, 112.37°).
The gon, or centesimal degree, is l/400th of the
full angle, a right angle being 100 gons.
The gon is the generally accepted "surveyor's
angle ", and is also used in aircraft navigation.
A gon is divided decimally either into 100
centesimal minutes, each minute into 100 centesimal
seconds,
1^ = 100c; lc = 100°°,
or into centigons and milligons,
1 gon = 100 cgon = 1 000 mgon.
Other names for the gon are grad and grade.

Section 12.2 Units of Measurement
411
Radian measure must be used in certain formulas in
mathematics and physics to avoid errors or the need for additional
scale factors. Otherwise, radians and degrees are at the user's
discretion.
2n rad = 360°
1 rad * 57.3°
To convert Sn/5 rad to degrees, we have
Sn J Sn J 360°
Trad=-rad'2^ad = 108-
To give 58.26° in degrees, minutes, and seconds, we have
60"
0.26° = 0.26 •— = 15.6'; 0.6' = 0.6-60" = 36",
58.26° = 58° 15' 36" . •
Units are easily converted with an electronic calculator with
modes for degrees and radians; results in radians are
approximate unless a highly sophisticated calculator giving results
in n is used.
In radian measure, the word radian(s) or its abbreviation,
rad, is generally omitted, whereas the symbol °, representing
degree(s), is always written.
Solid Angles
The measure of a solid angle (£2) is the area it cuts out on the
unit sphere, that is, a sphere whose radius is unity:
A solid angle need not have any particular shape; it is defined
by the conical surface of the nappe enclosing it. Thus, solid
angles of equal magnitude may have entirely different
shapes.

412
Chapter 12 ELEMENTARY GEOMETRY
Steradian
(sr)
The SI unit of measurement of solid angles is the steradian
(sr), which is a solid angle with its vertex in the center of a
sphere and which cuts off from the spherical surface an area
equal to the square of the radius of the sphere. The full sphere
represents a spherical angle of 4n steradians.
\ Area = r2
Area = r2
The main importance of the solid angle is to measure, in
physics, the fraction of the total emission of radiation from a
source in space, whether an electric lightbulb, a radioactive
preparation, or the Sun.
The radius of the Earth is approximately 6 370 km;
consequently, the area of the surface it presents toward the Sun is
K- 63702 km2. The radius of the Earth's orbit around the Sun
- the average distance between the Sun and Earth, or the
astronomical unit (AU) - is some 150 million km.
Thus, the solid angle occupied by the Earth is
Q = —■ & 5.6 • 10~9 steradian,
1502 • 1012
and the share of the total energy - light and heat - irradiated
by the Sun that is received by the Earth is the ratio of Q. to the
4rc steradians of a full sphere:
Q
K
6 3702
4rc 4 • n • 1502 • 1012
406
900
10"9 * 0.45-10
that is, less than one two-thousand-millionth of the total
energy emitted by the Sun, but this suffices to sustain us.
2 • 6370 km
Earth
Sun
" 15106 ^
km

Section 12.3 Euclidean Construction
413
12.3 Euclidean Construction
EVCLIDIS
ELEMENTORVM
SEX
LIBRI FRIORE S
DEMON* THAT I
nb
HENRICO COETSIO
MATHEJ JLECTOF.E
•-.—-—r-ff-■<■-- "
inn tm **t*6t0m***
Title page from Henric Coets,
Euclidis Elementorum ...
(1705), intended for self-
instruction. Henric Coets
was a mathematics teacher at
the University of Leiden, The
Netherlands.
COETS Henric
(end 17th century-1730)
Construction in Euclid's plane geometry is founded on the use
of an unmarked straightedge and a pair of compasses.
To construct a line segment equal to a given segment AB,
draw a line Z, longer than AB, using a modern compass) we
thus deviate from the classical method which assumed the
compass to be collapsible.
4
B
I
X
Choose a point X on Z and set the compasses for radius AB. Use
X as a center and describe an arc intersecting line Z at Y.
The segment XY is equal to AB.
To construct the midpoint of a segment AB, use the compasses
to describe about each of the points A and B circles of equal
radius, greater than half the length of AB:
The line through the intersections of the arcs intersects AB at
its midpoint and is perpendicular to AB.
The center of a given circle is located at the intersection of
perpendiculars raised from the midpoint of arbitrary chords.
To construct an angle equal to a given angle ABC, draw a
ray OY.
Ar^ X
o With B as center, describe an arc of any convenient radius
intersecting BA and BC at D and E, respectively.
o With the same radius, and using O as center, describe
an arc XS, where S is at the point of intersection with OY.
o Using S as center and with the radius equal to the distance
DE, describe an arc that intersects arc XS; call the point of
intersection Q.
o Draw a ray from O through Q.
Z O is equal to Z B.

414
Chapter 12 ELEMENTARY GEOMETRY
To construct the bisector of an arbitrary angle ABC, use B as a
center and any convenient radius to describe arcs intersecting
BA and BC; call the points of intersection X and Y. Using X
and Y as centers of arcs of any convenient radius, describe
intersecting arcs; call the point of intersection Z.
Ray BZ is the bisector.
To raise the perpendicular to a line Z, at a point A on Z, bisect the
straight angle whose vertex is at A.
/\Z
I
X1 A
Bisector AZ is perpendicular to Z at A.
To construct a perpendicular from a point B, outside a line Z,
use B as a center and any convenient radius to describe arcs
that intersect Z; call the points of intersection X and Y.
Is
^
X
^
"7
-/
\/ 7
Use X and Y as centers and any convenient radius to describe
intersecting arcs; call the point of intersection Z.
Line BZ is perpendicular to Z.
To construct the parallel through a given point P to a given
line Z, draw through P any line t that intersects Z and make
angle /i a corresponding angle to /2.

Section 12.3 Euclidean Construction
415
To construct an isosceles triangle with the base angles double
418 *ne s*ze °f *ne third angle, divide a line segment AB at a point
D in the golden ratio. Centered at A, describe a circle of radius
AB. Set off chord BC = AD.
To construct a 60° angle, start by describing an arc of arbitrary
radius about endpoint A of the straight line AB; the arc
intersects AB at C. With the radius unchanged, describe an
arc about C. The arcs intersect at D.
Angle BAD is 60°, because triangle ADC is equilateral and
equiangular.
Regular Polygons
Mathematicians in ancient Greece could construct polygons
with 3, 4, 5, and 15 sides using only an unmarked straightedge
and a pair of compasses, as described in Euclid's Elements.
As all angles can be bisected, polygons with even multiples of
the above numbers of sides are also constructive, a fact known
to Euclid.
The sides of a regular hexagon inscribed in a circle are equal
to the radius of the circle; copying consecutive chords - with
the length of the radius - around the circle will give the sides
and vertices of the regular hexagon.
Using the straightedge to connect alternate vertices of the
hexagon produces an equilateral triangle.
The corners of a square inscribed in a circle are formed by the
intersections with the periphery of two mutually perpendicular
diameters of the circle.
To construct a regular pentagon, start with an isosceles
triangle ACD whose base angles ACD and ADC are double the
size of angle CAD (see the top of this page).
Bisect the base angles. Using CD as the radius, describe two
circles centered at C and Z>, respectively, intersecting the
bisectors of the base angles; call these points of intersection B
and E, respectively. Construct the regular pentagon ABCDE
by joining points A and B, points B and C, points D and E, and
points E and A.
6-gon
3-gon
4-gon
5-gon
E

416
Chapter 12 ELEMENTARY GEOMETRY
10-gon
p. 415
15-gon
BARROW Isaac
(1630 -1677)
The regular pentagon may be constructed more easily by
connecting every other corner of a regular decagon.
To construct a regular decagon draw an isosceles triangle
whose base angles are double the size of the third angle as
previously described. Copy the chord BC around the circle -
with end and starting points of consecutive chords coinciding.
A regular pentadecagon, or 15-gon, may be constructed by
inscribing in a circle an equilateral triangle ABC and a
regular pentagon AEFGH.
Since arc AF is 2/5 and arc AB is 1/3 of the circumference of
the circle, arc BF is the difference 2/5 - 1/3 = 1/15 and,
consequently, chord BF is the length of one side of a regular
pentadecagon inscribed in the circle. Copying this chord
around the circle - with end and starting points of consecutive
chords coinciding - completes the construction of the sought
polygon.
For more than 2000 years, it seemed that the pentadecagon and
its even multiples were the acme of Euclidean construction of
regular polygons. The English mathematician Isaac Barrow
expressed himself with a prudent "as yet" at the end of the
Fourth Book in his "conveniently portable" edition of Euclid,
wisely keeping an eye open toward future achievements.
EVCLIVs
ELEMENTS.
The Whole
FIFTEEN BOOKS
Compeuctioujly Demonfiratei
By Mn$AAr Eamlow , Fellow ofrw"
nity College wcAMBfclDGE.
Ani Tra*Jl4ted out of the Latin.
JJicroct.
TkSeconil EAitiw, vci*y carefully corrected.
Printed f(£Wi/fyW Hnjcy a>an& 3\ la
LittUBritMn. MDCLXXXVL
PROP. xy/.
In a circleghen AEBC to inscribe a quindecagpne
(or ffieenfidedfgure) equilateral and equiangular*
»Infcnbe an equilateral pentagone AEFGH
in the circle given, and*>alfo an equilateral
triangle ABC, then I fay BF is the fide of the
quindecagone required*
For the arch ABc is-J or tJ of that
periphery, whereof AF is for rt: therefore the
remaining part BF is tt of the periphery. And
therefore the quindecagone, whofefide is BF,
is equilateral; but it is equiangular alfo,*
became all the angles infill on equal arches of a
circle, whereof every one ij- of the whole
circumference. Therefore, &c.
Scholium.
A circle is C4>8,i6,&c by 6,4and 9,1.
geometrically/3,6,i2,&c. byi^and 9,1.
divided into ^ 5,10,20,8:0. by 1 r ,4.and 9,1.
parts, c 1.5,30*60,8«:. by 16,4 and 9,1.
Any other wayofdividingthe circumference
into any parts given is as yet unknown,
wherefore inche conftrudtion of ordinate figures we
are forced to have recourfe to meckan/lk arti»
fices, concerning which jou may confult the
Writers of Practical Geometry.
The End of the Fourth Boo^

Section 12.3 Euclidean Construction
417
GAUSS Carl Friedrich
(1777 -1855)
17-gon
DISQUISITIONES
ARITHMETICAE
ATCTOKl
D. CAROLO FRIDERICO GAVS9
LIPS I A E
1« eoMxiiixi afto Gkxk. Flxibchx*,,**
I I O I.
In 1796, Carl Friedrich Gauss in his first scientific
publication demonstrated that regular rc-gons can be constructed
with an unmarked straightedge and compasses if
n = 2lpip2- ... -Pk,
where I is any positive integer or zero, and the pi's are distinct
primes of the form
22S+ 1,
where s is an integer.
The key numbers are the primes of this special form. The first
few (s = 0, 1, 2, 3, and 4) are:
3 5-17 257 65 537
On the other hand, s = 5, 6, and 7 give composite numbers and
do not give us new constructible regular polygons.
There are several methods to construct a regular heptadecagon,
or 17-gon, besides the one suggested by Gauss. One of the
simpler methods, due to H. W. Richmond, is shown here.
Describe a circle on the line segment AOB, with O as center:
L
Disquisitiones arithmeticae (1801)
is one of the great standard works
of number theory. The text is
primarily concerned with prime
numbers, which were shown to
cut into geometry and other
branches of mathematics in the
most surprising way; it includes
Gauss's proof of the constructibil-
ity of the 17-gon and other regular
polygons.
O
O
O
O
Draw the perpendicular radius OC.
Bisect the perpendicular OC, and repeat the bisection to
produce OD = i OC.
Draw line AD.
Bisect angle ODA twice to make angle ODE = j ODA.
Draw a perpendicular to DE at D; bisect the generated right
angle to obtain angle EDF = 45°.
Describe a circle with AF as diameter to intersect OC at G.
Describe a circle with E as its center and EG as its radius;
this circle intersects diameter AOB at if4 and Hq.
Raise perpendiculars to AOB at if4 and Hq.
These perpendiculars intersect the periphery of the original
circle at points P4 and Pq.
P4 and Pq are the 4th and 6th vertices of a regular 17-gon,
with A = Pi; bisect arc P4PQ to obtain P5.
Copy chord P4P5 around the circle to complete the regular
17-gon.

418
Chapter 12 ELEMENTARY GEOMETRY
The Golden Section
A line segment that is divided into two segments, a greater a
and a smaller b such that the length of a + b is to a as a is to 6, is
divided in the golden section or the golden ratio.
h
a
We have the relation
a + b
a
a
b '
a
b
= t; t - -r- 1 = 0
a
b
a
Golden Number: p. 287
Divine Section
(sectio divina)
PACIOLI Luca
(1445 -1514)
whose positive solution for — is
a _ V5 + 1
b " 2
V5-1
The golden section is also known as the divine section after its
Latin appellation sectio divina, which was first used by the
Franciscan monk and mathematician Luca Pacioli in his
work De divina proportione, published in 1509 in Venice.
The golden section was, for about 2500 years, a recognized
aesthetic guide in art, even a decree absolute, governing the
shape and disposition of drawings and paintings, and in
architecture, where the facades of many public edifices were
proportioned according to this ratio.
The golden section has nowadays lost most of its appeal to
architects and artists.
A golden section may be constructed in several ways; we
choose the following.
Given: on a straight line RS, a line segment AB of length a.
p. 435
A E B
o Construct at B a perpendicular BD of length a.
o Bisect AB at E; with E as center and ED as radius, describe
an arc that intersects RS at C. By the Pythagorean theorem,
the radius here is \5—.
We now have
AB
a
V5- 1
AC |(1 + V5)
The line segment AC is divided in the golden ratio by the
point B.

Section 12.3 Euclidean Construction
419
Golden Triangle A golden triangle is an isosceles triangle whose side is to its
base in the golden ratio; its angles are 72°, 72°, 36°. If a base
angle is bisected, two isosceles triangles are generated, one of
which is a new golden triangle. Using this method, new
golden triangles may be generated ad infinitum. If arcs are
drawn through the vertices as shown below, we obtain an
p. 580 equiangular, or logarithmic, spiral. This is one of an infinite
family of logarithmic spirals having various pitches.
A rectangle whose long side is in a golden ratio to the short
Golden Rectangle side is a golden rectangle:
(l+/i5)a
2a
By removing from a golden rectangle a square one side of
which coincides with the shorter side of the original rectangle,
we generate another golden rectangle. By this method, new
golden rectangles may be generated ad infinitum. If arcs are
drawn between nonadjacent corners of the square as shown
below, we obtain an approximation of a logarithmic spiral; the
true logarithmic spiral does not touch the sides of the
rectangles.
(l+/i5)a
<—2a —>
<-2b-+
Popular books on "the wonders and joys of mathematics" often
haul out the golden rectangle as unique in producing a
logarithmic spiral. However, any rectangle can be split into
parts, one of which is similar to the whole rectangle. The
splitting can be continued ad infinitum, giving rise to a
logarithmic spiral analogous to the one of the golden rectangle
but of a different pitch.
r
i
i

Chapter 12 ELEMENTARY GEOMETRY
The five-pointed star described by the diagonals of a regular
pentagon is a pentagram, or pentacle, whose central part is
another pentagon, the diagonals of which form yet another
pentagon, etc., in an endless succession of pentagons and
pentagrams. The triangles forming the points of the stars are
golden triangles. Each of the ratios AC/CD,AB/BC, and
AD/AE is equal to o(l + V5).
1 E
ik
The Pythagoreans used the pentagram as a secret
identification emblem; later it became a trademark of alchemists and,
perhaps because of its repeating properties, a sigu of the occult.
According to legend, it was used by Doctor Faustus to exorcize
Mephistopheles.
A progression of diminishing pentagons and pentagrams,
linking vertices together as shown below, is Pythagoras's lute;
the diagram is replete with lines in golden ratio:
A hexagon is a polygon with six sides. The internal angles of
a regular hexagon measure 120°.
By extending the sides of a regular hexagon to the points of
intersection, we obtain a hexagram, that is, the six-pointed
Star of David.
o A

Section 12.3 Euclidean Construction
421
Three Impossible Problems
Three geometrical problems have attracted the interest of
mathematicians and geometers since ancient times:
o Duplication of the cube - to construct a cube whose volume
shall be twice that of a given cube.
o Squaring the circle - to construct a square whose area
shall be equal to the area of a given circle.
o Trisection of an arbitrary angle - to construct an angle that
is exactly one-third of a given angle.
These tasks are "impossible problems" since they cannot be
solved by Euclidean methods, using exclusively an unmarked
straightedge and a pair of compasses. This has not, however,
restrained mathematicians from producing solutions, exact
and approximate, by other means.
Theorem of Thales: p. 430
Pythagorean Theorem: p. 435
62 + /12 = d?
a2 + h2
- *2
(P
+ C2 =
(a + 6)2
> h = yab
Duplication of the Cube
The Delian problem, or duplication of the cube, is the task of
constructing a cube whose volume shall be twice that of a given
cube. One of the many legends surrounding the problem tells
how the people of Athens, in 430 B.C., appealed to the Oracle of
Delos for advice to rid their city of the plague and were
instructed to double the size of their cubic altar to Apollo, whose
wrath they had supposedly incurred.
If we assume the edge of the altar (cube) to be a units in length,
we are faced with the task of solving the equation jc3 = 2 a3,
and - with x = 6 - to construct a length 6 = a y2 .
If a rational solution of this problem could be found, where
a and 6 are integers with no factor in common, we would have
that 63 = 2 a3 is an even integer; since the cube of an even
integer is even, 6 must also be even, say, 6 = 2p, which gives
63 = 8p3 = 2a3,and
a3 = 4p3,
that is, a3 is an even integer and, consequently, a is also even,
and a and 6 have a common factor, 2, which contradicts the
original hypothesis. We can conclude that the polynomial
jc3 - 2 = 0 is irreducible over the field of rational numbers;
thus, is an irrational number.
The barrier to the construction of a length, by the exclusive use
of an unmarked straightedge and a pair of compasses, is not
the irrationality of a number as such. Given any constructible
lengths l\ and h we can construct their sum and difference,
and by proportionality their product. We can also construct yl,
given I: construct a semicircle (as to the left) with a = I, 6 =1; if
a = 2 and 6 = 1, we can construct the irrational V2. This is,
however, all that we can achieve to construct lengths.
"V2 cannot be constructed from rational numbers by square
roots, and so, as proven in 1837 by P. L. Wanzel, the Delian
problem is not solvable.

422
Chapter 12 ELEMENTARY GEOMETRY
HIPPOCRATES of Chios
(2nd half of 5th century B.C.)
ARCHYTAS of Tarentum
(about 375 B.C.)
Many of the ancient Greeks concerned themselves with the
solution of the Delian problem. Hippocrates of Chios and
Archytas of Tarentum (now Taranto, Italy) were the first to
demonstrate that the problem could be reduced to the insertion
of two mean proportionals x and y between the edge length a of
a cube and twice its length,
a I x = x I y - y /2a ;
x is then the sought edge length of the doubled cube.
Squaring the Circle
Squaring the circle is the expression used to describe various
procedures for the geometric construction of a square that is
equal in area to a circle of given radius.
The problem of constructing a square whose area shall be
exactly equal to that of a given circle,
a2 = n - r2>
becomes a task of constructing a line segment whose length is
proportional to the square root of tc,
a = r
Vrc
Theorem of Thales: p. 430
Pythagorean Theorem: p. 435
K2 +x2
= y2
1+x2 =
^ X
y2 +z2
= (7t+l)2 J
= Vrc
This would be possible if we could describe a semicircle on a
diameter compounded of two line segments whose lengths are
1, and n units, respectively. A perpendicular raised at the
point where the two sections of the diameter meet will have
the length yn, but classical constructions can only produce
algebraic numbers. Since n is a transcendental number, the
segment n cannot be constructed, so this approach fails.
Squaring Lunes
Hippocrates of Chios constructed, in the 5th century B.C.,
crescent-shaped figures - lunes - whose area, in contrast to
that of the circle, may be squared; thus, there exists a square
whose area is exactly equal to that of a lune. That particular
lunes could be squared gave hope to those engaged in trying to
square the circle. The lune to the left is one of several lunes
that may be squared.
o Make a square AB CD with sides s.
o Form a lune ADC from intersecting lines of one circle
centered at B with radius s and another circle centered at E
(midpoint of AC) with radius half the length of AC.

Section 12.3 Euclidean Construction
423
Pythagorean Theorem: p. 435
Area of a Circle: p. 442
Areaiune ADC — Areasemicircle ADC ~~ An^circle segment AFC •
Radiusgemicircle ADC = 2 s ^¾ and thus:
Areasemicircle ADC = 4" ^ s
Areacircie segment AFC = Areacjrcie sector ABC — Areatriangle ABC
Circle sector ABC is one-fourth of the circle; thus,
1 s2
Areacircle sector ABC = "J K S ; Areatnangle ABC = 1£ •
We find
which corresponds to the area of a right triangle with legs s, an
area equal to that of a square with sides -rs *v2, which can be
constructed.
p. 511
Trisecting an Angle
Certain specific angles can be easily trisected, but there is
no general procedure that permits the construction, with
Euclidean tools, of an angle that is exactly one-third of a
given, arbitrary angle.
A 90° angle can be trisected, since a 30° angle is easily
constructed by using only a straightedge and a pair of compasses.
Since an obtuse, or larger, angle may always be split into one
or more right angles plus an acute angle, the problem of
trisecting an arbitrary angle can be reduced, without loss of
validity of the thesis, to the task of trisecting an acute angle. It
is then sufficient that we find just one angle that cannot be
trisected according to Euclid for the thesis to be proved.
We assume that the angle 3 0 = 60° is to be trisected, and jump
ahead to information given in Chapter 13 and use the
trigonometric identity
cos 3 0=4 cos3 6-3 cos 6,
where we insert cos 3 6= 60° = -r, and substitute
y
cos 6 = —
3 cos 6 =
3y
4 cos3 6 = ^
and obtain
y3-3y-l = 0,
an irreducible cubic equation, whose roots cannot be
constructed using Euclidean methods.
Therefore, cos 20° cannot be constructed with straightedge and
compasses and, hence, the angle 20° is not constructive.
The angle 60° cannot be trisected, and the thesis is proved.

424
Chapter 12 ELEMENTARY GEOMETRY
ARCHIMEDES
(287 - 212 B.C.)
Non-Classical Trisection
An interesting method of trisecting an arbitrary angle by non-
classical means is attributed to Archimedes.
Let AOB be the angle to be trisected.
o Construct a circle with O as its center and with any
convenient length as its radius.
o Next, construct a secant through B that intersects the
extension of diameter AC produced at a point D and the
circle at E, such that DE is equal to the radius of the circle.
The point D must be found by trial and error; a straightedge
is rotated around B until ED = EO.
o The angle ADB is then one-third of AOB.
For proof, we observe that base angles ODE and DOE of
isosceles triangle DEO are equal and, similarly, base angles OEB
and OBE of isosceles triangle EOB are equal. OEB is an
external angle to triangle OED and, consequently, equal to the
sum of angles ODE and DOE. Analogously, AOB is the sum of
OBD and ODB.
Consequently, ADB is the sought one-third of AOB.
As a curiosity, we present the simple instrument below:
V£
a
a
Let POQ be the angle to be trisected. Place the tool with its edge
Hi on the vertex O of the given angle, one leg of the angle
passing through the corner point A and the other leg as a
tangent to the quadrant at D.
Since the triangles OABy OCB, and OCD are congruent, the
angle POQ, or AOD, is trisected by the points B and C on the
extensions of tool edges H\ and #2> respectively.
p. 579
Trisecting an angle is also accomplished by use of the spiral of
Archimedes.

12.4 Theorems and Formulas
425
4fc
ftiiftfti
HFM
i '^mMmms^M^^ivM^
IJlvVf**^
ELEMEN
TA
GEOMETRlAB EX EVCLIDB
fmgulari prudcntu ccl/cfl* i
lo*Mir Vogdiii profcfforc
Kathenutico m fcko*
U Victmcnfi.
ARITHMETlCAB PRACTI*
ci per Gcorgium Prara
bachium Mdtbcnuti*
cum.
Cum pr/jfccioMc PWfypi M**
Idnthom.
i +-~m
ttfi»>K.d\
(3,'Cx.ct «g.
E V C L I D I S
ELEMENTORVM GEO-
METR1COKVM LIBR1 SEX
conuerfiin latinum fcrmoncm a
loach. Camcrario.
Q_V IDVS ADIECT.t 5VNT
mum priorum librorucn Demonfttationet, atque
eJitx in gratiam & vnlitatcm fluiliofoium Ma-
themaucesin Acad. Lipf. A
MauritioSteinmctz Gcrsb.
Mcdicinx Liccntiato.
L I P S I J£
Anno
M.
U X X V I I.
Johann Vbgelin's (? - 1549) Elementa
geometriae ex Evclide, first published in
1528, was a popular textbook and served as
a model for many later texts. This title page
is from the 1536 edition (Wittenberg).
Joachim Camerarius's (1500 - 1574) Evclidis ...
was first published without proofs and mainly
intended as a Greek primer. This title page is
from the 1577 edition (Leipzig), where proofs in
Latin had been added to many of the theorems.
This section is principally concerned with theorems from
Euclidean geometry. Strict proofs are given only for a limited
number of theorems; those given are not always of Euclid.
z
A
a, b, c,...
a,/3, /, ...
e
A
B
C
d
h
hc
m
mc
Notations
angle
triangle
sides of a triangle,
quadrilateral, etc.
peripheral angles
central angle of a
area
area of the base of i
circumference of a
segment of a line;
circle
i solid
circle
diagonal
of a polygon; diameter of a
circle
height (altitude)
height to side c
median
median to side c
P
P
r
r
R
s
S
SL
Sj<
t
tc
V
perimeter
- p = semiperimeter
radius of any circle
radius of inscribed
circle
radius of
circumscribed circle
arc
surface area
(boundary) of a solid
lateral surface area
of a solid
total surface area of
a solid
bisector
bisector to side c
volume

426
Chapter 12 ELEMENTARY GEOMETRY
12.41 Plane Geometry
Angles
complementary angles: p. 388 If two angles are complements of equal angles, then the two
angles are equal.
supplementary angles: p. 388 If two angles are supplements of equal angles, then the two
angles are equal.
Vertical Angles Vertical angles are equal:
a=al; /J=/J'
a + P = 180° (definition of a straight line);
a' +p= 180°.
a is supplementary to /?; a* is supplementary to /5.
Since supplements of the same angle are equal, cc= a* .
The pairs of alternate interior, alternate exterior,
corresponding, and opposite angles, formed by a transversal intersecting
a pair of parallel lines, are equal:
71
«1
«1
71
«1
«2
= «2;
= 72;
= «2;
= 72;
= 7i;
= 72;
A =/½
A = ¾
A = A
£1 = £2
A = A
^2 = ¾
Why?
Alternate Interior Angles
Alternate Exterior Angles
Corresponding Angles
Opposite Angles
alternate interior angles
alternate exterior angles
corresponding angles
opposite angles

Section 12.4 Theorems and Formulas (Plane Geometry)
427
Congruent and Similar Triangles
Two triangles are congruent (=) if the following elements of
one triangle are equal to the corresponding parts of the other
triangle:
a
a
a
a'
a = a1
6 = 6'
a- a1
Two sides and
the included angle
are equal.
a'
a = a1
6 = 6'
c = c'
Three sides
are equal.
a'
a = a1
a= a'
p=p
One side and
adji
are
oining angles
equal.
Two triangles are similar (~) if any of the following
conditions are satisfied:
a
a
a1
a b
a' 61
a- a1
Two corresponding
are proportional;
angles are equal.
sides
included
a1
a b c
a ' b ' c '
Three sides are
proportional.
a
a
a
Two angles
are equal.

428
Chapter 12 ELEMENTARY GEOMETRY
Base Angles of an Isosceles Triangle
The base angles of an isosceles triangle are equal.
Why? A ABB' = A AB lB by two sides and included angle being
equal, so Z B = ZB\
Sum of Angles of Triangle
The sum of the angles of a triangle is 180°.
Why? ABC is an arbitrary triangle; DE is parallel to BC and has a
point coinciding with point A .
ZDAB + ZBAC + ZEAC = 180°, or, since alternate angles
Z DAB =Z ABC and Z EAC = Z ACB,
ZABC + ZBAC + ZACB = 180° .
External and Opposite Angles of a Triangle
An external angle of a triangle equals the sum of the two
opposite interior angles.
P + r
Why? The sum of the angles of a triangle is 180°, as is the angle of a
straight line.

Section 12.4 Theorems and Formulas (Plane Geometry)
429
Angles Subtended by the Same Arc
If a central angle and a peripheral angle are subtended by the
same arc, then the central angle is twice as large as the
peripheral angle.
Why? We distinguish three cases:
1. The center of the circle, O, lies on one of the sides of the
peripheral angle.
2. O lies between the sides of the peripheral angle.
3. O lies outside the sides of the peripheral angle.
Case 1
Case 2
Case 3
We start with Case 1 and the observation that
2fi+x = 180° =a + x;
so the peripheral angle is half the central one.
By drawing diameters AD2 and AD3, Case 2 and Case 3 are
reduced to Case 1.
a+x+2y= 180°
x + 2(P+y)= 180°
a = 2/3

430 Chapter 12 ELEMENTARY GEOMETRY
From the above we may also deduce that the peripheral angle
depends only on the intercepted arc:
Angles inscribed in a circle and subtended by the same
arc are equal.
a=/?
Looking at the symmetric case, with the intercepted arc equal
to a semicircle, we see that the peripheral angle is 90°, and we
have arrived at the theorem of Thales, named for the
mathematician Thales of Miletus:
Angles inscribed in a semicircle are right angles.
Quadrilateral in a Circle
Opposite angles of a quadrilateral inscribed in a circle are
supplementary.
a+y= p+8= 180'
p. 429 Why? This is a corollary of the theorem stating that the central angle
is twice as large as the peripheral angle if the two angles are
subtended by the same arc.
Theorem of Thales
THALES of Miletus
(c. 624 - 547 B.C.)

Section 12.4 Theorems and Formulas (Plane Geometry)
431
Angle Subtended by a Chord of a Circle vs.
Angle Formed between Chord and Tangent
The angle formed between a tangent to a circle and a chord
equals the inscribed angle on the other side of the chord and
subtended by the chord.
ax = a
Why? We realize the above by using a limiting argument based
on the previous theorem. As C moves toward D, T tends to
the tangent while a"1 = a" at all points and a1 tends to a'
while a tends to a:
Ellipsoid Reflectors
Radii from the foci of an ellipse to a point on the periphery
make equal angles with the tangent to the ellipse at that point.
This fact is the basis of ellipsoid reflectors; when emitted at
one focus of an ellipsoid, light rays and beams of sound are
reflected in such a manner that they converge into the other
focus.

432 Chapter 12 ELEMENTARY GEOMETRY
Paraboloid Reflectors
The angle between a tangent to a parabola and a radius from
the focus to the tangent point is equal to the angle between the
tangent and a line parallel to the axis of the parabola that
intersects the parabola at the tangent point.
P jT\a S
vf^~^ cf
vOr I \
/fa]
axis f I ^ [-
\F
This fact forms the basis of paraboloid reflectors; light rays
and beams of sound emitted from the focus of a paraboloid
reflector are reflected as a parallel beam. Conversely, a
parallel beam received by a paraboloid reflector is converged
into its focus; this is the principle behind the reflecting
telescope, where the image of the object appears at the focus of
the paraboloid, and parabolic satellite disks.
Concurrent Points and Lines
Intersection of Bisectors of a Triangle
The bisectors of the angles of a triangle are concurrent, and
intersect at the center of an inscribed circle:
Circumcenter of a Triangle
Normals from the midpoints of the sides of a triangle are
concurrent, and intersect at the circumcenter, the center of a
circumscribed circle:

Section 12.4 Theorems and Formulas (Plane Geometry)
The 9-Point Circle
The midpoints of the three sides, the base points of the three
heights, and the midpoints of the line segments between the
corners of a triangle and the intersection of the heights are on
a circle:
This fact is surprising and is, like the following theorem, one
of the high points of triangle geometry.
Euler's Line
In every triangle, the intersection of the medians, the
intersection of the heights, and the center of the circumscribed
circle are on a straight line.
Intersection of Medians of a Triangle
The medians of a triangle are concurrent, intersecting at a
point which is two-thirds of the distance from any vertex to the
midpoint of the opposite side; this is the center of gravity of the
triangle:
c

434
Chapter 12 ELEMENTARY GEOMETRY
DESARGUES Girard
(1591 -1661)
PONCELET Jean Victor
(1788 -1867)
Dual Theorem
Dual Elements
Principle of Duality
Dual Theorems of Projective Geometry
The French mathematician and engineer Girard Desargues
is the founder of the formal study of projective geometry. His
Brouillon projet ... (1639) and other works of fundamental
importance for the theory of projective geometry were, however,
largely ignored until the 19th century when another French
mathematician and engineer, Poncelet, revived the study of
projective geometry.
A dual theorem (or simply dual) of projective geometry is a
theorem that can be obtained from another by interchanging
the dual elements - lines and points (vertices). The principle
of duality of projective geometry is that if one of two dual
theorems is true, the other theorem is also true.
Desargues's Theorem
Collinear Points
Pascal's Theorem
Brianchon's Theorem
The lines joining opposite
vertices concur at P.
Desargues's Theorem: If the lines joining corresponding vertices of two triangles are
concurrent (that is, they meet in one point), then the
intersections of corresponding sides are collinear (that is, they lie on a
line). The dual is also true.
Pascal's Theorem:
PASCAL Blaise
(1623-1662)
Brianchon's Theorem:
BRIANCHON Charles Julien
(1783-1864)
If a hexagon is inscribed in a conic section, the three points of
intersection of pairs of opposite sides are collinear.
This theorem was first published in Essay pour les coniques
("Study of Conies") by the French mathematician-physicist-
philosopher Blaise Pascal in 1639, who had been inspired by
Desargues's recent work.
If a hexagon is circumscribed about a conic section, the lines of
intersection of pairs of opposite vertices are concurrent.
This theorem is named after the French artillery officer and
geometer Brianchon, who described it around 1806.
Brianchon's theorem is the dual of Pascal's theorem, and vice
versa.

Section 12.4 Theorems and Formulas (Plane Geometry)
435
PYTHAGORAS of Samos
(c. 580 - c. 500 B.C.)
The Pythagorean Theorem
In a right triangle, the sum of the squares of the lengths of the
legs is equal to the square of the length of the hypotenuse.
a2 +b2 = c2
a
The Pythagorean theorem, or Pythagoras's theorem - perhaps
the most renowned of all mathematical theorems - was known
by the Babylonians as far back as 2000 B.C., that is, about 1500
years before the time of Pythagoras and the Pythagoreans.
Classical Proof Consider the following figure - as Euclid did about 300 B.C.:
We make the following four assumptions:
o On a given straight line, it is possible to construct a
square; in Euclid's Elements, the proof of this fact
immediately preceded that of the Pythagorean theorem.
The existence of squares requires the Euclidean parallel
postulate.
o The area of a rectangle is the product of two adjacent sides.
o The area of a triangle is half the product of the height and
the base.
o Two triangles are congruent when they have two sides and
the included angle equal.
Let ABC be a triangle with BAC a right angle.
Let there be described on BC the square BDEC, on BA the square
ABFG, and on AC the square AHKC.

436
Chapter 12 ELEMENTARY GEOMETRY
Bhaskara
(1114 -c. 1185)
Bhaskara's Proof
Through A, let AL be drawn parallel to BD and CE, and draw
AD and AE.
The angles CBF and DBA are equal, for they are obtained
when the angle ABC is added to either of the equal (right)
angles DBC or FBA.
Since BF = BA and BC = BD, and the angles CBF and DBA are
equal, the triangles CBF and DBA are congruent.
The height of the triangle DBA - with respect to the base BD -
is DL (which is equal to BM). The height of the triangle CBF
- with respect to the base FB - is FG (which is equal to AB).
The area of the triangle DBA is half the area of the rectangle
BDLM (same base, BD, and same height, BM). Likewise, the
area of the triangle CBF is half the area of the rectangle ABFG
(same base, BF, and same height, AB).
As shown above, CBF = DBA, and consequently the area of the
square ABFG equals the area of the rectangle BDLM.
By an argument corresponding to the above, we can also show
that the area of the square AHKC equals the area of the
rectangle CMLE.
The area of square BDEC is composed of the rectangles BDLM
and CMLE, whose areas we have shown to equal the squares
ABFG and AHKC, respectively. Therefore, the sum of the
areas of the squares ABFG and AHKC equals the area of the
square BDEC. QED
There are many other verifications of the Pythagorean
theorem; the following geometrical proof, given about 1150 by
the Indian mathematician Bhaskara, provides a visual
feeling of correctness.
Assume that the area of a square is the square of its side, and
that the area of a right-angled triangle is half the product 'of its
smaller sides.
Inscribe a square in another square so that each corner of the
inscribed square coincides with a point of one side of the
circumscribed square.
The circumscribed square contains four right-angled
triangles (shaded areas) whose hypotenuse, c, is a side of the
inscribed square, and whose smaller sides are a and b.
The total area of the four triangles is
A ab
4—= 2ab.
The area of the circumscribed square whose side is (a + b) may
be expressed as (a + b)2, or as the sum of the area of the
inscribed square and the total area of the four right-angled
triangles; thus:
and we have
(a+ 6)2 = 2ab+c2
a2 + 2a b + b2 = 2ab + c2,
a2 + b2 = c2 .
QED

Section 12.4 Theorems and Formulas (Plane Geometry)
437
to
\)
fIrte Quicf^ and the (Dead
One pleasant summer evening Jred invited a group of fellow pure
mathematicians to a garden party. Chatting about surjections,
cohomology rings, unknotting conjecture, immersion of projective
spaces, and other relaxing topics, Andy suddenly remarl^ed:
"you've got a pretty tall flagpole, Jred. I bet it's about 10
meters high."
"Actually, if I remember right, it's 14."
They all got quite excited about this nontrivial problem. The
wagers came fast.
Jred borrowed a measuring rod from a neighbor.
As the mathematicians were about three-quarters of the way to the
top of the flagpole, climbing up on each other's shoulders, Jred's
young daughter Anna came home. When she found out what they
were up to, she blurted:
"Haven't you heard of the Pythagorean theorem?"
The heap of mathematicians tumbled to the ground.
x
This is what Anna did:
She measured the double string of halyard that exceeded the
length of the flagpole, and found the excess to be 1 meter. The
double string just reached the level ground when tightened at a
point 5 meters from the foot of the flagpole.
Letting the height of the flagpole be x meters:
x2 + 52 = (x + 1)2
x = 12 meters •

438
Chapter 12 ELEMENTARY GEOMETRY
The clue to proofs for theorems of geometry usually lies in
finding similar triangles and, with right-angled triangles,
the use of the Pythagorean theorem.
Diagonals and Sides of a Parallelogram
■/g><
d^ + d22 = 2(a2 + b2)
The diagonals bisect each other.
PTOLEMAEUS Claudius
(c. 100 - c. 168)
Ptolemy's Theorem
The sum of the products of the two pairs of opposite sides of a
convex quadrilateral inscribed in a circle is equal to the
product of the lengths of the diagonals.
—7lc ABxCD+ADxBC = ACxBD
The theorem is named after the mathematician, astronomer,
and geographer Ptolemy of Alexandria.
Quadrilateral About a Circle
a + c = b +d
(For proof, note that the radii of the
circle join the points of contact.)
Chord Theorem
ab = cd

Section 12.4 Theorems and Formulas (Plane Geometry)
439
Secant Theorem
a(a + b) = c(c + d)
Why? With similar triangles, we have
a
a + b c + d
Secant-Tangent Theorem
cf. p. 431
t2 = a (a + b) * *
a
Why? Again, with similar triangles:
t
i_t a
+ b ~ t '
Height, Median, and Bisector of a General Triangle
cf. Heron's formula: p. 444
hn =
2 Vp (p - a) (p -b)(p - c)
wherep = «■ (a + b + c).

440
Chapter 12 ELEMENTARY GEOMETRY
Radii of Circumscribed and Inscribed Circles
Cyclic Quadrilateral
Square
P_ 1 Ua c + d b) (a d + b c) (a b + c d)
" 4 \ (p-a)(p-b)(p-c)(p-d)
wherep = « (a + & + c + d) ; cf. Heron's formula: p. 444.
R =
a^2
r = 2 a
General Triangle
Equilateral Triangle
Right Triangle
r =
R =
2 ' -^Triangle
a + b + c
a b c
4 • ^Triangle
_2fi_
r =
R =
aV3
6
a V3
a b
r =
R =
a + b + c
hypotenuse
Circle
Curvilinear Figures
The principle of finding curve lengths will be discussed in the
chapters dealing with calculus.
C = 2nr = nd
Arc of a Circle
y/^rad^
anr
S=l80- = er
Ellipse
P * 2
K^sJ 2 ^2 +
b2)

Section 12.4 Theorems and Formulas (Plane Geometry)
441
Areas
Areas of Rectilinear Figures
Rectangle
The area of a rectangle is the product of two
adjacent sides,
A = ab ,
which is evident from the drawing below, where b
rows, each of a unit squares, fill the rectangle:
Square
Rhomboid
or Parallelogram
6 = 3
, ,*,«.*. .w.\ . .*.w .www .ww.wwv l.1 ..1 l * * W\"
a
A square may be considered a special case of a
rectangle with all sides equal. It has the area
A = a-a = a2 .
The area of a rhomboid (parallelogram), is the
product of the length of one side and the
corresponding height; for a rhomboid with sides a and b
and corresponding heights ha and h^, we have
A = a - ha = b - hb,
as depicted for A = a • ha in the illustration below,
where a right-angled triangle has been
transferred from one end of the rhomboid to the other,
transforming the rhomboid into a rectangle:
A
yfjjij
J»1I!I
a
bA
4
11
"a
U
a
Rhombus
Triangle
A rhombus is a rhomboid with all sides equal; the
area is obtained in the same way as for the
rhomboid.
The area of a triangle with sides a, 6, and c and
the corresponding heights ha , hby and hc is half
the products of base and height,
A = — aha = 2 bhb = 2 c hc ;
a triangle may always be considered one-half of a
corresponding rhomboid.

442
Chapter 12 ELEMENTARY GEOMETRY
Trapezium The area of a trapezium (trapezoid) is half the
(U.K.) product of the sum of the parallel sides times its
Trapezoid height; we have for a trapezium/trapezoid with
(U.S.) parallel sides a and 6, and height h,
A = 2 (& + b) h .
To demonstrate the correctness of this statement,
rotate the quadrilateral 180° and place the upside-
down replica beside the original to transform the
trapezium/trapezoid into a rhomboid of twice the
area:
a
a
Areas of Curvilinear Figures
The principle of finding areas of curvilinear figures will be
discussed in the chapters dealing with calculus.
Circle
Ellipse
A = nr2
A = nab
Sector of a Circle
Segment of a Circle
/oTRdy
A = 2 sr
"360s"
Or2
A = o (sr-ah)
Segment of a Parabola
a
A = g ab

Section 12.4 Theorems and Formulas (Plane Geometry)
443
Problems
We return to rectilinear figures.
An isosceles triangle has a perimeter of 12 units and a height
of 4 units with respect to the base.
Find the legs, base, and area of the triangle.
With the base of the triangle 26, we have, with Pythagoras,
a = Vl6 + 62
2a + 26 = 12
>
a = Vl6 + 62
a = 6-6
>
Vl6 + 62 =6-6; 16 + 62 = 36-126 + 62
126 = 20,
. 5
6=3;
1 ^
26 = 83
>
a = 43
A-»*.|.4.f
= 6r square units.
Find the length of a side of an equilateral triangle with an
area equal to that of a square whose side is 5 cm.
Let the side of the triangle be 2 a , and h its height.
4 a2 = h2 + a2
h =a V3 .
A = ah => a2V3
= 25
a = 5
4r—
V27
2a =
4_
V27
10 • -t— cm.

444
Chapter 12 ELEMENTARY GEOMETRY
A square is inscribed in a right triangle with legs 7 and 14 cm;
one side of the square is coincident with the hypotenuse and its
two remaining corners are on the legs of the triangle.
Find the area of the square.
Let the side of the square be x.
-tui«^
2x
We have, with Pythagoras,
AC = Vl42+ 72 = V245 = 7^/5 cm.
Since triangles ACB, ADG, and EFC are similar, we have
x
and
CF = 2x ; AG = j ,
— + x + 2x = 3 £• x = V245 = 7^5
x = 2a/5 ;
and the area of the square
x2 = (2V5) 2 = 20cm2.
Heron's Formula
HERON
(Probably 1st century AD)
Area of a Triangle
Heron of Alexandria - one of the greatest engineers and
inventors of antiquity - pioneered the science of applied
mathematics and was a prolific writer and compiler of the
knowledge of mathematics and engineering of his time.
His name is attached to formulas for the calculation of the
areas of triangles, quadrilaterals, and certain regular
polygons of higher orders, although - as is often the case - they
may have been known by some of his predecessors.
The area of a triangle can be calculated if the lengths of its
three sides are known,
A = Vp (p -a) (p -b) (p -c),
where a, 6, and c are the sides of the triangle, 2p = a + 6+cis the
perimeter, and p is called the semiperimeter.

Section 12.4 Theorems and Formulas (Plane Geometry)
445
Proof The height h9 perpendicular to the base c of the triangle, can be
calculated by the Pythagorean theorem:
Arbitrary Quadrilateral
Cyclic Quadrilateral
h2
h2
= a2 - (c - ci)2 = b2 -
-ci2
a2 = b2 + c2 - 2 c ci
b2 + c2 - a2
Cl " 2c
,o (b2 + c2 - a2\
=b-[ 2c
2
J
4c2/i2= 4 62c2-(62 + c2-a2)2
= [2 b c + (b2 + c2 - a2)] • [2 6 c - (62 + c2 - a2)]
= [(6 +C)2_a2] . [a2-(6 -c)2]
= (a + b + c) (- a + b + c) (a - b + c) (a + b - c)
where we introduce
a + 6 + c = 2p -a + 6 + c = 2(p-a)
a-6+c = 2(p-b)
a + 6-c = 2(p-c).
4c2h2 = 16p(p-a)(p-6)(p-c)
and find the area
A = — = Vp (p -a)(p - b) (p - c) .
QED
Heron has also given formulas for the areas of quadrilaterals
whose sides and angles are known.
The area of an arbitrary quadrilateral is
A = V(p - a) (p - b) (p - c) (p - d) -abed cos2 a ,
where
a, 6, c, <i are the sides of the quadrilateral
2p = the perimeter, a + b + c + <i
a = half the sum of two opposite angles.
For a cyclic quadrilateral - a quadrilateral that can be
inscribed in a circle - the formula simplifies to
A = V(p - a) (p - 6) (p - c) (p - d) .

446
Chapter 12 ELEMENTARY GEOMETRY
12.42 Solid Geometry
Volumes

Parallelepipeds
The volume V of a rectangular parallelepiped
with edges a, b, c is the product of its edge lengths,
V = a • b - c .
A parallelepiped with edges a,b,c can be
subdivided into c layers with b rows of a unit cubes
each:
lO
II
W
/
/////
//////
A- '
1
/
/
/
/
K.
X>
a = 6
The bottom of the rectangular parallelepiped has
an area ab, and the height is c; if we call the base
area B and the height h, we have the volume
V = Bh.
Cube
Analogously, the volume of a cube with a = b = c is
V = a ■ a • a
aK
Oblique
Parallel-
Epipeds
For the volume of an oblique parallelepiped, we
can extend the method of reasoning used for
oblique parallelograms, or use Cavalieri's
theorem to find
V = Bh.
Cavalieri's
Theorem
CAVALIERI Bonaventura
(1598 -1647)
Prisms,
Cylinders
Solids of equal height have equal volumes if
sections parallel to and equidistant from their
bases have equal areas, that is,
V = B-h.
The theorem is named for the Italian physicist
and mathematician Bonaventura Cavalieri.
The volume of a prism is equal to the product of the
base area B and the perpendicular height h,
V = Bh.

Section 12.4 Theorems and Formulas (Solid Geometry)
447
This also applies for all cylinders, which can be
considered as prisms with an infinite number of
lateral faces.
The solids shown in the above drawing all have
the same base area at both ends, and the same
sectional area at all levels between the ends. By
Cavalieri's theorem, their volumes are equal.
Pyramids, The volume of all pyramids - or cones, which
Cones may be looked upon as pyramids with an infinite
number of lateral faces - is one-third of the
product of the base area B and the corresponding
height h,
V = \-Bh.
A triangular prism can be split into three
pyramids of equal volume by two plane sections.

448
Chapter 12 ELEMENTARY GEOMETRY
The Five Regular Polyhedra
To construct a vertex of a regular polyhedron, two conditions
must be met:
- a minimum of three planes of regular polygons is required;
- the sum of the angles of the polygonal vertices must be less
than a full angle of 360°.
Regular
polygon
Triangle
Square
Pentagon
Hexagon
No. of
vertex
polygons
3
4
5
6
3
4
3
4
3
angle
60°
90c
108°
120c
5um of
ingles
180°
240°
300°
360°
270°
360°
324°
432°
360°
Regular
polyhedron
Tetrahedron
Octahedron
Icosahedron
None
Cube (hexahedron)
None
Dodecahedron
None
None
It is not possible to construct a regular polyhedron with regular
hexagons or polygons of higher order.
Thus, there are just five regular polyhedra.
The Euler Characteristic
DESCARTES Rene
(1596-1659)
EULER Leonhard
(1707-1783)
If a polyhedron has V vertices, F faces, and E edges and is
topologically equivalent to the sphere, one will always find
V + F-E = 2,
where the 2 is the Euler characteristic of the polyhedron. The
formula - originally devised by Rene Descartes and later
resurrected by Leonhard Euler - is often referred to as Euler's
polyhedron formula. We will see how the Euler characteristic
for other shapes can be calculated by breaking them into
polyhedra.
Consider a polyhedron with the least possible number of
faces - a tetrahedron:

Section 12.4 Theorems and Formulas (Solid Geometry)
449
Topologically, the tetrahedron is equivalent to a sphere. On a
sphere, denote the vertices of the tetrahedron with dots, the
edges with arcs; the areas between the arcs correspond to the
faces of the tetrahedron:
We may displace the vertices, edges, and faces so that they are
all exposed from one direction:
We have 4 dots (V), 4 fields (F), and 6 arcs (E), which gives
4y+ 4jp-6e = 2 .
An arc can be added in two ways inside a triangular field:
o from a dot to an opposite arc
- giving one new dot, one new field, and two new arcs;
o between adjacent arcs
- forming two new dots, one new field, and three new arcs.
Thus, whichever way an arc is added, the sum of the increase
in dots and fields (V + F) equals the increase in arcs (E) and,
consequently, V + F - E remains equal to 2.
The formula V + F - E = 2 is true for any polyhedron, as long
as its interior is a topological ball:
For the polyhedron at the right we must count the 6 faces in the
well at the top as well as the 5 quadrilateral faces that ring that
well, giving a total of 17 faces; thus,
20^+17^-35^ = 2.

450
Chapter 12 ELEMENTARY GEOMETRY
Genus: p. 379
As we have shown, the Euler characteristic is 2 for a surface
topologically equivalent to the sphere. If topologically
equivalent to a torus the surface has the Euler characteristic 0.
Topologically equivalent to a torus, the prism with a hole
straight through its center has an Euler characteristic of 0,
2(V+2(V-40£ = 0.
To classify a polyhedron topologically, we may determine its
Euler characteristic, and to do so we make the paradoxical use
of counting the number of vertices, faces, and edges, none of
which need to be preserved under topological equivalence.
For two-sided surfaces the Euler characteristic is
V + F-E =2-2p,
where p is the genus of the surface, while for one-sided
surfaces
V + F-E = 2-p,
withp the genus as before.
Selection of Formulas from Solid Geometry
Many of the following formulas will be further discussed in
later chapters, concerned with calculus.
V = Volume
S = Lateral surface
Regular Polyhedra
R = Radius of circumscribed
sphere
r = Radius of inscribed sphere
Tetrahedron
Cube
2R
V =
s =
R =
r =
a3V2
12
a2V3
4
12
V = a3
S
R
6a2
aV3
a
2
Octahedron
2r
V =
a3V2
S = 2a2V3
aV2
R =
r =
2
aV(3
6
Dodecahedron
Icosahedron

Section 12.4 Theorems and Formulas (Solid Geometry)
451
Dodecahedron
Icosahedron
R
a3 (15 + 7 V5)
S = 3a2V5(5 + 2V5)
a(l+V5) V3
a ^ /50 +
22 V5
5 a3 r-
V = ^-(3 + V5)
S = 5a2V3
i? = y ^2 (5 + V5)
4
-f-vP
3V5
Prisms
General Prism
B
V= Bh
Parallelepiped
V=Bh
Rectangular
Parallelepiped
/^
1 k
: v
1 \
/
k
a
V = a&c
d = Va2 + 62 + c2
Pyramids
v = j £/i
y= i a (#! + b2 + VBi £2)

452
Chapter 12 ELEMENTARY GEOMETRY
Cylinders
V = Bh
Cones
/5:::¾¾¾¾
vN
::¾
V = nr2h
S = 2nr(r + h)
General Cone
Right Circular Cone
V=^Bh
V = \nr2h
S = nr(r + s)
Lateral surface
area = nr s
Frustum of a
Right Circular Cone
V = 3 n h (ri2 + ri r2 + r22)
S = n [ri2 + s (ri + r2) + r22]
Lateral surface
area = ns (ri + r2)
Spheres and Spherical Segments
S = 4nr2
V = ^nhOr^ + h2)
V = | n h (3 r^ + 3 r22 + h2)
S = 2nrh
S = 2nrh

Section 12.4 Theorems and Formulas (Solid Geometry)
453
Spherical Triangle
n _ 2 ^degrees _ 2 f
^ "" n r 1 qq ~" r ^radians >
where E is the spherical excess;
this is shown here for a semi-
lune but is true for all spherical
triangles.
Torus
V = 2n2r2R
S = 4n2rR
Problems
The corners of a cube are truncated so as to present a solid
bounded by six regular octagons and eight equilateral
triangles.
Determine the area and volume of the truncated cube.

454
Chapter 12 ELEMENTARY GEOMETRY
p. 444
Let the side of the octagon be a and the cube edge s.
We then have
a = s(<2-l)
a2 = s2(3-2V2)
a3 = s3(sV2-7)
s = a + 2
V2
= a (V2 + l)
The area A of the truncated cube is the sum of six octagon areas
Ag and eight areas A3 of the equilateral triangles.
An octagon consists of eight isosceles triangles with base a
and height s /2; their aggregate area is
6A8
= 6-(8---a \s) = 12a2(V2+l)
With the perimeter 2 p of an equilateral triangle of side a,
2p = Sa
3a
a
P =
(p-a) = 2
we have, with Heron, the aggregate area of eight triangles,
^f¥W- 2°2^
8 A3 = 8
and the total area
A = 6A8 + 8A3 = a2(l2V2+12 + 2V3) ,
or, in terms of s,
A = s2 (l2 V2 + 12 + 2 V3) (3-2 V2)
* 5.565 s2.
The volume V of the truncated cube equals the volume Vq of the
intact cube less the aggregate volume 8V3 of eight right-angled
corner pyramids,
V0 = S3
« 1 1 /"a V «*^2
^ = ^3-2(^ =a3~
= s3 I (5 <2 - 7) y[2 = |s3(l0-7^),
and the total volume is
V=V0- 8V3
0.966 s3

Section 12.4 Theorems and Formulas (Solid Geometry)
455
Some chemical compounds which normally crystallize in a
cubic structure will sometimes form "twin cubes". These have
a space diagonal in common, around which one cube has
rotated through 60° relative to the other.
Determine the total area and volume of the twin cube.
The twin cube may be regarded as a conventional cube, each of
whose six faces carries a ridge formed by an edge and corner
of the other cube. Each ridge adds three right triangles to the
surface area of the twin cube,
two side flanks, with legs s and — ;
s s
one end face, with legs — and —,
corresponding to an aggregate area
A+ = 2
A
V
2S
s 1 i £ £ «2_ — 2
2 ' 2 ' 2
8 8
From this must be deducted the area A_ of the base surface on
which the ridge is sitting, an isosceles triangle with
and
the base — y2
the legs - V5,
from
V
fs\2
k2] +S'
The area of this base surface can be calculated in two ways,
o" base x height or by Heron's formula; we will demonstrate
both.
The height of the triangle is

456
Chapter 12 ELEMENTARY GEOMETRY
Heron's formula gives
2p = | (V5 + VB + V5) ; p = I (2 a/5 + V2)
p-a = I (2^5 + V2 - 2V2) = J (2^5 - V2)
p-b = p-c =|(2V5 + a/2- 2 V5) = I a/2;
A_ = ^a/(2a/5 + V2) (2V5 - V2) (V2)2
«s2 3 2
= 616 =8S •
The total area of the twin cube is
The total volume of the twin cube is
S +6'2'2'2'3S =14s •
* * *
Answer to pie problem from page 447:
The pie has radius R; the inner circle has radius r.
2 m 2 R^
nrz = kRz —nrz : r = —-— .

457
Chapter
13
TRIGONOMETRY
Page
13.0 Scope and History 458
13.1 Fundamental Trigonometric Functions 470
13.2 Inverse Trigonometric Functions 477
13.3 Solving Triangles 478
13.4 Graphs, Domains, Ranges 502
13.5 Trigonometric Identities 507
13.6 Trigonometric Equations 516
13.7 Limits 528
*** 31 Harmonic Analysis 907
***
Indicates cross-reference
•

458 Chapter 13 TRIGONOMETRY
13.0 Scope and History
Trigonometry developed from the study of right-angled tri-
Greek, trigonon, "triangle"; angles by applying their relations of sides and angles to the
-metria, "measurement" study of similar triangles.
Plane Trigonometry Plane trigonometry relates exclusively to triangles in the two
dimensions of the plane, but has applications in nearly every
field of physics, such as radiation, the propagation of light and
sound, alternating current, and all other periodic phenomena.
Spherical Trigonometry Spherical trigonometry is concerned with triangles on the
surfaces of spheres; its uses are more limited, the principal
applications being to astronomy and long-range navigation.
The primary use of trigonometry is for operations of
surveying, cartography, astronomy, and navigation. Modern
mathematics has extended the uses of trigonometric functions
far beyond a simple study of triangles to make trigonometry
indispensable in many other areas.
Although made up of Greek phonemes, the term
"trigonometry" is actually not a native Greek word. To the best of
our knowledge, the term was invented by the German math-
PITISCUS Bartholomaeus ematician and astronomer Bartholomaeus Pitiscus and first
(1561-1613) appeared in his work Trigonometria sive de solutione trian-
gularum tractatus brevis et perspicius ..., first published in
1595; it was revised in 1600 and published as Trigonometria
sive de dimensione triangulae.
Trigonometric functions, which are today dimensionless
numbers describing ratios of the sides of right-angled
triangles, have a varied history.
The ancients had no clear conception of trigonometric
functions as elements of a geometric science, but they knew how to
use them to advantage in the furtherance of their chosen
professions.
The old Egyptians looked upon trigonometric functions as
features of similar triangles which were useful in land
surveying and when building pyramids; the old Babylonian
astronomers related trigonometric functions to arcs of circles
and to the lengths of the chords subtending the arcs.
The ancient Greeks who took over from the Babylonians as
astronomers developed trigonometry into an ordered science
and constructed tables of chords expressed in terms of 60th
"small parts" of the radius for every 2 degree from 0° to 180° of
the central angle.
Centuries later, trigonometric functions acquired a geometric
interpretation when they came to be looked upon as the lengths
of specific line segments related to the central angle.

Section 13.0 Scope and History
459
V
V* >
- <(i
^4
[^
s«
■ft ^
\
\
;:jSii
, . „»artholom&'i x
F'mfct Grun !>erdenii5.
Trigonometric
Problematvai \ftRIOKVT>
nenxve [
Geotlarticorum,
Altimetncdrum,
Geodmphicorum,
Cncmiomcorurn, e£
A ftronomicorvim:
V^/s
■*4-*i
#^
r*
,>*#< J)VftV
■*v
:^
^

460
Chapter 13 TRIGONOMETRY
In a circle whose radius is 1 unit of length - a unit circle -
the radius OP is assumed to rotate in a counterclockwise
direction. When moving from its initial position OA to the
terminal position OP, it sweeps the angle AOP = 0.
tangens
tangens
arcus
The tangents to the circle at points A and B meet the extension
of the radius OP at E and F, respectively. The line segments
PC, perpendicular to OA, and PD, at right angles to OB, help
to form triangles OCP, OAE, and OBF, which are similar,
angles being equal as marked.
The tangent GPH at P is divided into two segments: PG is
equal to the tangent line segment AE, and PH is equal to the
cotangent segment BF.
With OA = OB = OP = 1, and observing that the angle BOF is
the complementary angle of AOP = 0, we have the following
equalities:
AE
tangens 0 = ttt = AE = PG
radius
arcus
= OP= 1
= AP
" OA
BF
cotangens 0 = jr^ = BF = PH
ARYABHATA Kusumapura
(476 -c. 550)
sinus 6
PC
OP
oc
= PC = OD secans 0
OE
OA
OF
= OE = OG
cosinus 0 = op = OC = PD cosecans 0 = ^~ = OF = OH
The origin of the term "sinus" - our "sine" - is Indian. The
Hindu mathematician and astronomer Aryabhata the Elder
called it ardha-jya ("half-chord"), later abbreviated to jya
("chord"). Arab translators turned this phonetically into jiba
- a meaningless word in Arabic - and, according to Arabic
practice of omitting the vowels in writing, wrote it jb.

Section 13.0 Scope and History
461
Arabic-to-Latin translators, with no knowledge of the
Sanskrit origin, assumed jb to be an abbreviation of jaib,
which is good Arabic for "cove", "bay", "bulge", "bosom".
Gerard of Cremona, who translated the Almagest in the late
12th century, substituted for jaib its Latin equivalent sinus.
The names of the other trigonometric functions have less
complicated interpretations. In the figure, tangens of 6 is
represented by the length of the tangent segment (AE),
secans of 6 by the secant segment (OE). The cotangens and
cosecans are, of course, the tangens and secans of the
complementary angle.
The tangens, cotangens, secans, and cosecans functions have
been known by various names; the present names - tangens
from Latin tangere, "to touch", and secans from secare, "to
cut" - were introduced in 1583 by the Dane Thomas Fincke.
The names cosinus and cotangens, suggested in 1620 by the
English mathematician and astronomer Edmund Gunter,
replace the older terms sinus complementi and tangens
complementi - sinus and tangens of the complementary
angles.
Masters of the Angle
There are problems in the Rhind Papyrus from about 1650 B.C.
that describe the measurement of face slope angles of
pyramids, involving what is today known as the cotangent
function of the dihedral angle at the base of the pyramid.
A
h
aA
A Babylonian clay tablet from around 1900 - 1600 B.C. contains,
in cuneiform script, a table of the secant functions of fifteen
angles between 30° and 45°. The wealth of astronomical data
collected by the Babylonians was passed on to the ancient
Greeks and gave rise to spherical trigonometry, essential in
astronomy and navigation.
THALES The Greek philosopher Thales of Miletos is said to have made
(c 625 - c. 547 B.C.) use of similar triangles to determine the height of the Cheops
pyramid by comparing the length of its shadow with the length
of the shadow cast by a rod of known length; he also
determined the distances offshore of ships at sea.
GERARD of Cremona
(c. 1114-1187)
FINCKE Thomas
(1561 -1656)
GUNTER Edmund
(1581 -1626)

Chapter 13 TRIGONOMETRY
ARISTARCHOS
(c. 310 - 250 B.C.)
Aristarchus of Samos, mathematician and astronomer, and
the first man to propound a heliocentric theory of the Universe
- eighteen centuries before Copernicus - made an attempt to
compare the distances from Earth to the Sun and to the Moon.
Moon
Earth
Sun
ERATOSTHENES
(c. 276 -194 B.C.)
HIPPARCHOS
(? - after 127 B.C.)
Although his reasoning was perfectly sound, the instrument he
used to determine the angle of sight between the Sun and the
half Moon failed him by faulty calibration - he found the
distance to the Sun to be about 18 to 20 times that to the Moon,
instead of the correct figure of approximately 390 times.
The first known attempt to calculate the circumference of the
Earth was made by Eratosthenes of Alexandria. Having
heard that, at the summer solstice, the Sun was at zenith at
Syene - modern Aswan - he decided to determine the height
of the Sun also at Alexandria. From his measurements he
deduced, correctly, that the distance between Alexandria and
Syene must equal 1/50th of the Earth's circumference, but all
his other data were inaccurate or pure guesswork.
Apart from delivering severe criticism of the disdain for
fundamental data and the rather cavalier treatment of them
evinced by Eratosthenes, Hipparchus of Nicea and Rhodes is
justly famous for his manifold contributions to astronomy;
his achievements include the determination of the length of the
lunar month to within one second of today's accepted value and
accurate calculations of the inclination of the ecliptic and of
the changes of the equinoxes.
As a mathematician, Hipparchus introduced to Greek science
the Babylonian method of dividing the circle into 360°.
It was common practice in ancient Greece, when solving
triangles, to assume them to be inscribed in a circle, their
sides being chords of the circle. Hipparchus is known to have
prepared extensive tables of chords at ^-degree intervals for all
central angles from 0.5° to 180°, calculated to three positions by
the Babylonian sexagesimal system.
The radius of the circle was divided into 60 equal segments,
called "small parts" used as unit of length; the first figure of
the chord length refers to the number of small parts needed to
make up the length of the chord, the second figure is the
number of 60ths of those parts, and the third figure is the
number of 3600ths.
Like all of Hipparchus's numerous works, except one, this
table is lost; knowledge of most of Hipparchus's work has
come down to us through Ptolemy.

Section 13.0 Scope and History
463
MENELAOS
(c. A.D. 100 )
PTOLEMAEUS Claudius
(c. 100 -c. 168)
Arabic,, al majisti, "the greatest";
Arabicized superlative form
of Greek megas, "great" and
magiste, "greatest"
••X.^
Gregor Reisch, Margarita phyl-
osophica (1496; 1503 reprint).
Escorted by Dame Astronomy,
Ptolemy is using a quadrant, an
instrument for measuring
altitudes. Ptolemy is wrongly
identified as one of the 15 Ptolemy
kings of Egypt (323 - 30 B.C.)
and, therefore, drawn wearing a
crown. In the left lower corner is
an armillary sphere (an ancient
instrument composed of rings
showing the positions of
important circles of the celestial sphere
(that is, the imaginary sphere
against which the celestial
bodies appear).
ABU AL-WAFA
(940-997 or 998)
umbra versa: " shadow
(ghost) behind arcus"
NASIR AD-DIN
(1201 -1274)
ADELARD of Bath
(active c. 1116-1142)
JOHN of Seville
ROBERT of Chester
PLATO of Tivoli
(c. 1100)
GERARD of Cremona
(c. 1114-1187)
Menelaus of Alexandria, in the first century A.D., wrote
a six-book treatise on chords - mentioned by Theon of
Alexandria, but now lost - and also made important
contributions to spherical trigonometry, providing later writers with
a basis for further development of the subject. It is interesting
to note that early achievements in trigonometry were
predominantly directed toward spherical trigonometry, in
response to the need for accurate information in astronomy.
A three-book work by Menelaus, called Sphaerica, is extant in
an Arabic translation and provides excellent information
about Greek development of trigonometry.
In the second century A.D., the great mathematician and
astronomer Ptolemy of Alexandria presented an extensive
treatise on all aspects of astronomy in his 13-book Syntaxis
mathematica ("Mathematic Collection"), also known as "the
greatest of the great", and commonly referred to as the
Almagest, from a Latin distortion of the Arabic al-majisti,
meaning "the greatest".
The first book of the Almagest contains tables of chords for all
arcs 0° to 180° at 0.5-degree intervals to at least five places of
decimals, believed to be the table prepared by Hipparchus and
mentioned by Theon. The tables are equivalent to a table of
sines for all central angles 0° to 90° at 15' intervals.
The Almagest also contains theorems corresponding to the
present-day law of sines and compound-angle and half-angle
identities.
That so much of early Greek work on astronomy has been lost
could be a result of the completeness and elegance of
presentation of Ptolemy's Almagest, making all earlier works appear
superfluous.
In the 9th to 14th centuries, trigonometry was further developed
by Arabic and Persian scholars. As with Babylonian and
Greek trigonometry, the incentive came from interest in its
application to astronomy.
Building further on Ptolemy's Almagest, in the 10th century,
the Persian Abu al-Wafa of Baghdad systematized theorems
and proofs of trigonometry and prepared extensive
trigonometric tables of sines and tangents for 15' intervals; he is
believed to have introduced the concept of the tangent
function - called umbra versa - and possibly also the secant
and the cosecant.
The first presentation of trigonometry as a science
independent of astronomy is credited to the Persian Nasir ad-Din
in the 13th century. In his writings we meet, also for the first
time, plane trigonometry as a discipline in its own right,
separate from spherical trigonometry.
In the 12th century, many Greek, Hebrew, and Arabic texts
were translated into Latin, often by Jewish translators
working in Spain. Notable translators were Adelard of Bath, John
of Seville, Robert of Chester, Plato of Tivoli, and Gerard of
Cremona - the translator of the Almagest into Latin.

464
Chapter 13 TRIGONOMETRY
FIBONACCI Leonardo
(c. U70 - c. 1250)
REGIOMONTANUS
(1436 -1476)
Leonardo Fibonacci, also known as Leonardo Pisano, became
acquainted with trigonometry during his extensive travels in
Arab countries; his newly acquired knowledge was presented
in Practica geometriae in 1220.
Around 1464 the German astronomer and mathematician
Regiomontanus, also known as Iohannes Molitoris, compiled
De triangulis omnimodis, a compendium of the trigonometry
of that time. After being printed in 1533, it became an
important medium for spreading knowledge of trigonometry
all over Europe.
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(kkguLui.Nohinbcrgz in aedibua Io.Pctrd.
AHHO CHRIST!
H. O. XXXllL
RH^ETICUS Georg Joachim
(1514-1574)
The German mathematician and astronomer Georg Joachim
Rhaeticus in 1542 published, separately, under the title De
lateribus et angulis triangulorum, the section of Copernicus's
De revolutionibus dealing with plane and spherical
trigonometry, with an added table of half-chords subtended by arcs of
a circle - that is, in effect, a table of sines, although both he and
Copernicus avoided the term sinus,
Rhaeticus's sine table differs from that of Copernicus in
two respects: its radius has been increased from 100 000 to
10 000 000, and the interval of the central angle is 11 instead
of 10'. The high values of the radii were used to avoid
fractions.
Rhaeticus's Canon doctrinae triangulorum, published at
Leipzig in 1551, was the first publication to give all six of the
fundamental trigonometric functions, and the first in which
they were defined as ratios of the lengths of the sides of a right
triangle and related directly to the angles - not, as before, as
lengths of line segments or ratios of an arc.
Rhaeticus managed to reduce by half the space occupied by his
tables by equating functions of angles greater than 45° with the
corresponding cofunctions of their complementary angles.

Section 13.0 Scope and History
465
OTHO Valentinus
(c. 1550 -1605)
FINCKE Thomas
(1561 -1656)
PITISCUS Bartholomaeus
(1561 -1613)
IKTRODUCTIO
JH ANAL T S I N
INFINITORUM.
JUC TO R £
LEONHARDOEULERO,
Pra/tfore S^m Biiolininit. \& jtejuUmutm-
ftriJitScUaturum P«ti.ovolit.iH4
Socio.
TOMUS PRIMUS
LAUSANNE.
Apd Uxrcum-Michaiiiu Bouiqjiit St Sodo*-
MDCCXL7UL
EULER Leonhard
(1707-1783)
After the publication of Doctrinae, Rhaeticus embarked upon
the project which was to occupy him and a number of
computing collaborators for at least 12 years; on the death of
Rhaeticus in 1574, Valentine Otho, a mathematics student at
Wittenberg, took over the enormous task of compiling the
manuscript for the Opus palatinum de triangulis, to consist of
a canon of sines for the range of 0° to 90° at 10" intervals, and
for every 1" within the ranges of 0° to 1° and 89° to 90°, all to 15
decimal places, and a complete canon of sines, tangents, and
secants to 10 or 15 decimal places.
The Opus palatinum was eventually published in 1596, but it
was marred by systematic errors in the tangents and secants
between 83° and 90°, later to be corrected by Bartholomaeus
Pitiscus.
The Danish physician and mathematician Thomas Fincke
published in 1583, at Basle, a 14-book treatise called Geo-
metriae rotundi... with discourses on plane and spherical
trigonometry, in which he introduced the terms tangens and
secans, and several new formulas, e.g., the law of tangents.
The German mathematician Bartholomaeus Pitiscus
published at Heidelberg, in 1595, a treatise named Trigonome-
tria: sive de solutione triangularum tractatus brevis et
perspicuus, noted for the first appearance in print of the word
trigonometry.
A revised version of the work appeared at Augsburg, in 1600,
as Trigonometriae sive de dimensione triangulae; it treats
all aspects of plane and spherical trigonometry: definitions of
functions, methods of solving triangles, trigonometric
identities, and tables for the sine, tangent, and secant
functions. Enlarged versions were published at Augsburg in
1609 and at Frankfurt/Main in 1614.
Pitiscus's works became valuable textbooks; English
translations appeared in 1614, French in 1619. The English text -
A Canon of Triangles: or the Tables of Sines, Tangents and
Secants, the Radius Assumed to be 100000 - is based on the
1595 Trigonometria and on trigonometric tables from later
publications by Pitiscus.
The transformation to modern trigonometry is to a great
extent attributed to work by the Swiss mathematician Leonhard
Euler, whose name is also firmly linked with fundamental
contributions to calculus and topology. For almost a hundred
years, Euler's Introductio in analysin infinitorum (1748) was
the dominating textbook of trigonometry and other
fundamental areas of mathematics.
The history of trigonometry, and of geometry in general, is
linked to technical advancement in instrument making; For
practical land measure, the ancient surveyors' simple
instruments - a knotted rope, a wooden rod, and a tool similar to an
ordinary carpenter's square - usually gave satisfactory
results. But we have also seen - in Aristarchus's calculations
of interplanetary distances - how an inaccurate instrument
can invalidate what is right in theory.

466
Chapter 13 TRIGONOMETRY
HULSIUS Levinus
(? -1606)
Born in Flanders; lived and
worked in The Netherlands,
England, and Germany.
Maker of fine instruments;
publisher and printer;
linguist and lexicographer; wrote
extensively on the
construction of geometrical
instruments.
(StfittXtamt
LEVINI HVLSII.
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■*- I I .Mil
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M. DC. IIIX.
The Swedish astronomer
Anders Celsius (1701 - 1744),
was one of the members of the
1736 expedition to Lapland to
measure the length of the
degree along the meridian.
The results of this expedition,
led by French mathematician
and astronomer Pierre-Louis
Maupertuis(1698 - 1759),
verified Isaac Newton's hypothesis
that the Earth is flattened at the
poles and thus resembles a
spheroid (p. 405) more than a
sphere.
"Celsius the Astronomer" Linoleum cut by Knut Erik Lindberg (1921-1988).
In his left hand, Celsius holds a sextant pointed at the North Star; in
his right hand is a thermometer.

Section 13.0 Scope and History
467
Wangled Angles
Eratosthenes, head librarian of the science library at
Alexandria - the largest in Antiquity - was also active as a
mathematician and a geographer.
As a geographer, he is best known for his Map of the World,
with a zero meridian and a "zero" parallel both passing
through Alexandria.
As a mathematician, he was known for his boasts that he had
solved the (unsolvable) Delian problem of doubling the cube.
The eminent Greek geographer Strabo was later to
characterize him as "a mathematician among geographers and a
geographer among mathematicians".
Around 230 B.C. Eratosthenes was told that at Syene in southern
Egypt the Sun was at zenith at the time of the summer solstice;
its rays were penetrating vertically down to the bottom of a deep
well on the Elephantine Island in the Nile without
illuminating the walls of the well.
He assumed that Syene was situated on the Tropic of Cancer
(the most northerly latitude at which the Sun can be seen at
zenith) and that it was located due south of Alexandria - it said
so on the world map that he had himself prepared. Actually,
Syene (now Aswan) is about 3 degrees east of Alexandria,
corresponding to a difference of 12 minutes of the clock in the
Sun's passing through the meridian - of no practical
importance for the measurements, but a fact.
At Alexandria, he measured the angle of incidence of the
Sun's rays with a scaphe - a hemispherical bowl with a needle-
shaped gnomon placed at its center. The inside surface of the
bowl is graduated to permit the length of the arc of shadow cast
by the gnomon to be read.
From the observed length of this shadow Erathostenes could
deduce that his baseline of measurement - the arc of the
Earth's surface between Alexandria and Syene - must occupy
l/50th of the Earth's circumference along the meridian.
Referring to his world map, it is open to some doubt what he
meant by "circumference" and "meridian", as the map shows
his world as a large flat mass of land floating in the Oceanus.
What the distance between Alexandria and Syene might be, he
did not know; nor did anybody else at the time. Without
recourse to trigonometric methods - which he certainly did not
have - it was impossible at his time to measure such distances.
Erathostenes had been told, however, by wayfarers and camel
drivers that one could go by caravan from Syene to Alexandria
in 25 days - or even in 20 days by fast camel. Erathostenes
therefore decided on a distance of 5000 stadia - divisible by
both 20 and 25 - though there were knowledgeable people who
held that the distance was 4530 stadia. It could be that both
parties were right - or both wrong - as there were at least seven
different lengths to choose from for a stadion; we have
no means today of knowing which one he used for his
calculations.

468 Chapter 13 TRIGONOMETRY
Having decided in favor of 5000 stadia, he calculated the
Earth's circumference as 50 x 5000 = 250 000 stadia. This figure
had the disadvantage, however, of not being divisible by 60,
disagreeing with Egyptian practice. So - as attested by
Hipparchus and Strabo - he added another 2000 stadia for
good measure, to bring the supposed circumference up to
252 000 stadia, which is divisible by 60.
Eratosthenes' assumed 5000 stadia was pure guesswork, and so
was his 250 000 stadia, not to speak of the 2000 extra stadia and
his reason for adding them, a swindle on top of his surmise.
The only thing that Eratosthenes actually knew was that the
distance between Syene and Alexandria occupied l/50th of the
Earth's circumference.
Eratosthenes' method of calculating the circumference of the
Earth is scientifically sound, but most of his data were
inaccurate or pure guesswork - if not worse. His contemporaries
recognized him for an eclectic and an opportunist, but many
later-day chroniclers have allowed themselves to be fooled by
the fact that his various guesses, mistakes, and errors largely
managed to cancel each other out.
Knowing the correct figure of the Earth's circumference to
be very near 40 000 km and the actual distance between
Alexandria and Syene/Aswan to be just under 800 km, they
make 5000 stadia correspond to 800 km, and so,
50 x 5000 stadia = 50 x 800 = 40 000 km.
Some people call this kind of science "fidelity to ancient
tradition".
Mystery Angles
© Jan Norrman, Central Board of
National Antiquities, Sweden
Ale's Stones. A stone ship ...

Section 13.0 Scope and History
469
[Roslund, Curt: See Bibliography]
Located on the southern coast of Sweden, Ale's Stones were
long thought to be only a stone ship raised to honor a Viking
chief named Ale (pronounced a as in "father", e as in pet).
If, in fact, just a stone ship, it is uniquely composed of
antipodal parabolas - other stone ships generally have sides
shaped in an arc of a circle. Though the stones are not in
perfect parabolic order today, investigations confirm an
original parabolic design; stones that had fallen were
reerected slightly out of line with the original design.
A captivating question now is whether this majestic
arrangement of stones might represent an astronomical instrument,
like the 4700-year-old construction at Stonehenge, near
Salisbury in Wiltshire, England. Among the supporting
testimonies are the parabolic outline and the stern of the ship
pointing toward the winter solstice, but crucial evidence of
remains from an era matching that of Stonehenge still falls
short; in fact, radiocarbon dating of charred wood found on the
site sets the age of that wood at about A.D. 400 - considerably
younger than the supposed age of Stonehenge.
Investigations and speculations will no doubt continue.
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Chapter 13 TRIGONOMETRY
13.1 Fundamental Trigonometric Functions
opp
The ratios of the lengths of any two sides in a right triangle
will depend on the magnitude of the acute angles of the
triangle; these ratios, of which there are six, are called
fundamental trigonometric functions of the angles.
Referred to the angle a in the triangle below, these functions
are:
opposite side
hypotenuse
read: "sine of a"
sine
sin a =
sinus
cosine
tangent
cos a =
tan a =
cotangent cot a =
secant
cosecant
sec a =
esc a =
adjacent side
hypotenuse
opposite side
adjacent side
adjacent side
opposite side
hypotenuse
adjacent side
hypotenuse
opposite side
cosinus
tang ens
cotangens
secans
cosecans
These definitions lead to the following relations between the
functions:
tan =
sin
cot =
cos
sec =
esc =
cos sin cos sin
The cotangent is the reciprocal of the tangent function, the
secant of the cosine, and the cosecant of the sine:
111
cot a =
tan a
sec a =
cos a
esc a =
sin a
Exponential Form
The fundamental trigonometric functions can be defined in
terms of exponential functions and the imaginary unit; thus,
xlX
e1* - e
-1 JC
>1*
+ e
—1 JC
sin x =
tan* =
2i
■i (eix
i-ix
)
COS* =
cot* =
i(eix + e~ix)
ki*
+ e
-1 X
>1 x
e1* - e
-1 X
where e denotes the base of the natural logarithms and i
denotes the imaginary unit.

Section 13.1 Fundamental Trigonometric Functions
Complementary Angles
Two angles whose sum is a right angle - 90° - are called
complementary angles; e.g., the acute angles of a right-
angled triangle:
a = 90° -p
p = 90° -a
Consequently, the sine of one of the acute angles is always
equal to the cosine of the other, complementary, angle; and
correspondingly for all other functions and cofunctions:
sin a = cos p tan a = cot p sec a = esc p
cos a = sin p cot a = tan p esc a = sec p
a+p= 90°

472
Chapter 13 TRIGONOMETRY
The Orthogonal Coordinate System
Origin
jc-axis, y-axis
The fundamental trigonometric functions, so far defined only
for acute angles, may be extended to apply for all angles with
the help of unit circles, of radius 1, centered at the origin of an
orthogonal coordinate system with two real-number axes, a
horizontal a>axis and a vertical y-axis, intersecting at right
angles at the origin.
Second Quadrant
y
i
4
3
2
1--
-4 -3 -2 -1
-1
-2
-3
Third Quadrant _^ -
First Quadrant
P (3, 2)
x
12 3 4
Fourth Quadrant
Abscissa
Ordinate
Radius
Positive Angle
Negative Angle
An arbitrary point on the coordinate plane is defined by its
perpendicular distances from each of the two axes: the
^-coordinate, or abscissa, is its horizontal distance from the
y-axis; its y-coordinate, or ordinate, is the vertical distance
from the x-axis. The point P in the graph above has the
abscissa = 2, and the ordinate = 3.
A ray issuing from the origin of the coordinate system is
known as a radius; the angle it forms with the positive x-axis
counts positive if produced by counterclockwise (CCW)
rotation and negative if caused by clockwise (CW) rotation of
the radius:
x
- 27C/3 rad
i
27C/3 rad
Khi/3 rad
x
x

Section 13.1 Fundamental Trigonometric Functions
473
Observing that the radius of a unit circle is r = 1, we have by
our previous definition that, for acute positive angles 0,
sin 0 = —
r
x
cos 0 = —
r
y
tan 0=-
x
cot e = -
y
and we may declare these equations to be definitions for all
possible values of 6:
y
X
X
-y
X
X
y
As an angle increases through all four quadrants, the
definitions remaining the same, the sign of the trigonometric
functions will change:
Sine
Cosine
Tangent
Cose
+
—
cant
t
+
—
Sec
J
—
—
:ant Cotai
+ -
+ +
igent
+
—

Chapter 13 TRIGONOMETRY
Values of Trigonometric Functions for Any Angle
X
First Quadrant: Acute Angles
sin 0 = — = y
1 J
x
cos 0 = — = x
tan 6 =-
x
CSC 0 = —
sec 0 = —
1
y
l
cot e = -
X
y
Second Quadrant: Obtuse Angles, Supplementary Angles
x
sin 0 = sin (180° - 0)
cos 0 = - cos (180° - 6)
tan 6 = - tan (180° - 6)
esc 6 = esc (180° - 6)
sec 0 = - sec (180° - 6)
cot 0 =- cot (180° - 6)
Third Quadrant: Reflex Angles
sin 6 = -sin (0-180°) esc 6
cos 6 = - cos (0- 180°) sec 6
tan0= tan (0-180°) cot 0
esc (0-180°)
sec (0-180°)
cot (0-180°)
Fourth Quadrant: Reflex Angles
sin 0 = - sin (360° - 0) esc 0
cos 0 = cos (360° - 0) sec 0
tan 0 = - tan (360° - 0) cot 0
esc (360° - 0)
sec (360° - 0)
cot (360° - 0)

Section 13.1 Fundamental Trigonometric Functions
475
Analogously, for any negative angle (- 0):
sin (-0) = - sin 0 esc (-0) = - esc 0
cos (-0) = cos 0 sec (-0) = sec 0
tan (-0) = -tan 0 cot (-0) = -cot 6
Finding All the Fundamental Trigonometric
Functions of an Angle
If the numerical value of one trigonometric function of an
angle is known, and we know in which quadrant the angle is
located, the remaining trigonometric functions can be
calculated.
24
• With cos 0 = - —, find the sine, tangent, and cotangent.
We have, with Pythagoras,
242 + x2 = 252 ; x = ±7
24
There are two solutions, with 6 in the 2nd or 3rd quadrant,
7 7 24
2nd quadrant: sin 6 = — tan 6 = -— cot 0 = — —
25 24 7
7 7 24
3rd quadrant: sin 6 = - — tan 6 = — cot 0 = -=-
25 24 7
Conversion Table
sin x =
COS* =
tan x =
cot* =
sine
—
± Vl - sin2 #
sin #
Vl - sin2 x
Vl - sin2 x
± — :
sin #
cosine
± Vl - cos2 #
Vl - COS2 #
+
COS #
COS X
H
Vl - cos2 #
tangent
tan #
+
V1 + tan2 x
1
+
V1 + tan2 x
1
tan x
cotangent
1
+
V 1 + cot2 X
cot X
V 1 + cot2 #
1
cot #

476
Chapter 13 TRIGONOMETRY
Values of Some Trigonometric Functions
radian: p. 410
Degrees
0
30
45
60
90
120
135
150
180
210
225
240
270
300
315
330
360
Radians
0
n
6
n
4
n
3
n
2
2n
3
3k
4
5 n
~6~
n
In
6
5 n
4
An
~3~
3k
2
5 n
3
In
4
UK
6
2k
sine
0
1
2
V2
2
V3
2
1
V3
2
V2
2
1
2
0
1
"2
V2
2
V3
2
- 1
V3
2
^2
2
1
"2
0
cosine
1
V3
2
V2
2
1
2
0
1
"2
^2
2
V3
2
- 1
V3
2
^2
2
1
"2
0
1
2
^2
2
V3
2
1
tangent
0
V3
3
1
V3
undefined
-V3
- 1
V3
3
0
V3
3
1
Vs
undefined
-V3
- 1
V3
3
0
cotangent
undefined
Vs
1
V3
3
0
V3
3
- 1
-V3
undefined
V3
1
V3
3
0
V3
3
- 1
-Vs
undefined
Calculators and Trigonometric Values
Tables of trigonometric functions have largely been replaced
by electronic calculators; as these generally do not possess
keys for the cotangent, secant, and cosecant functions, they
have to be computed with the reciprocal functions:
cot0 =
tan 6
sec 6=
cos 6
esc 6
sin 6

477
13.2 Inverse Trigonometric Functions
Arcus Function
An inverse trigonometric function, or arcus function, is the
reverse of a fundamental trigonometric function.
For instance,
sine of 30° is 0.5 => arc sine of 0.5 is 30° .
The inverse trigonometric functions are:
arc sine denoted arcsin
arc cosine
arc tangent
arc cotangent
arc secant
arc cosecant
arccos
arctan
arccot (or arcctg)
arcsec
arccsc (or arccosec)
Domains and Ranges: p. 339
p. 505
We read arcsin x as "The arc sine of x" or "The inverse sine
of %".
The practice of denoting inverse trigonometric functions by
sin-1 x, cos-1 x, etc., is discouraged because of possible
confusion with the expressions (sin x)~l and (cos *)_1, equivalent
to 1/sin x and 1/cos x.
For a trigonometric function to have an inverse which is also a
function, the domain of/*, and, as a consequence, the range of
the inverse, must be restricted. Domains and ranges of
fundamental and inverse trigonometric functions will be
discussed in Section 13.4.
Geometrically - with measurements in radians - the arc sine
function is the arc length of the unit circle for a given value of
a fundamental trigonometric function, as reflected in the
Latin expression arcus cuius sinus x est ("the arc whose sine
is x"), and thus accounting for the arc part of the names of
inverse trigonometric functions.
Find angles a and /3 of the triangle:
cos a =
vi
Vs
a = arccos -~- = 30°.
sin p =
vi
. V3
p = arcsin -~- = 60° .

478
Chapter 13 TRIGONOMETRY
13.3 Solving Triangles
Right-Angled Triangles
To solve a right-angled triangle we must know one side and
one acute angle, or two sides.
Find the diameter of a circle which circumscribes a right
triangle with one leg equal to 5 cm and the opposite angle 48°.
By the theorem of Thales, the hypotenuse is the diameter D of
the circle.
sin48c
_ JL n- 5
" D ' sin 48°
& 6.73 cm.
An isosceles triangle may be solved by using one of the two
right triangles formed when the angle between the equal sides
is bisected. Similarly, a regular polygon may be solved after
forming an isosceles triangle by connecting the center of the
polygon with the endpoints of one side:
In the above ra-gon, O is its center, R the radius of a
circumscribed circle, r the radius of an inscribed circle, and s the side
of the /z-gon. The central angle of the isosceles triangle is
3607/z. The central angle of each right triangle formed by
bisecting the isosceles triangle is = 1807n.
n
We have
sin
180c
n
s/2
R
180°
cos
n
r
1"
tan
180c
n
s/2
r
s
27

Section 13.3 Solving Triangles
479
ARCHIMEDES
(c. 287 - 212 B.C.)
Archimedes, eminent Greek mathematician, physicist, and
inventor/engineer of the 3rd century B.C., used circumscribed
and inscribed regular 96-gons to determine the numerical
value of k, though not by trigonometric means, to between 3 -rr-
1
and 3 —
We shall use circumscribed and inscribed regular polygons to
determine the numerical value of n between even closer
boundaries than those achieved by Archimedes.
Let the side of the circumscribed 180-gon be 2 a, that of the
inscribed polygon 2 6, and the radius of the circle r.
\
J
1°
^^
f
^K
- 26 ►
2a ■
We have
where
a
b
= r
= r
180-2
• tanl°
• sin 1°
180-26 < 2nr < 180 2a
180 • 2r • sin 1° < 2 nr < 180 • 2 r • tan V
180 sin 1° < n < 180 tan 1°
3.1414 < n < 3.1419 .

480
Chapter 13 TRIGONOMETRY
Arbitrary Triangles
Law of Sines
To solve a triangle when we know
two angles and one opposite side
PTOLEMAEUS Claudius
(c. 100 - c. 168)
Why?
or
two sides and an opposite angle,
we may use the law of sines, demonstrated about A.D. 150 by
Ptolemy of Alexandria:
sin A sin B sin C
a
B
The altitude h is
h = 6 sin A , and h = a sin B,
which gives
b sin A = a sin B
sin A sini?
a ~ b
In the same manner, it can be shown that
sin A sinC
a c
Find the length of the longest side of a triangle with angles
66.0° and 52.0°, if the shortest side is 35 cm.
The third angle is 62.0°, < 66.0° but > 52.0°. Thus, the longest
side must be opposite the angle 66.0° and the shortest side
opposite 52.0°.
Let the longest side be x cm.
By the law of sines,
x 35
sin 66.0° sin 52.0°
35 sin 66.0°
x =
sin 52.0(
?* 40.6 cm.
The Ambiguous Case
Inspector Sally Miles found the body of the intruder, electrocuted
when trying to inactivate the alarm system. In his hand was a
parched but still readable note:

Section 13.3 Solving Triangles
481
At the site of the old millstone, a triangle. Two sides: 15
and 10 yards; 3(f angle opposite the 10-yard side: corner
located at the center of the millstone. Third side: due east
of the millstone. Money: 5 feet down in a dry well at the
corner of the 10-yard side and the third side.
"Again, trigonometry to the fore," eiQilted the inspector. She
found the old millstone, read the compass, marf^ed off a triangle
with the given measurements, and dug a good 5-foot hole.
'But the money was not therel
a<Hgt to worry," said the inspector to her assistant, Jonathan 'Bell.
"This must Be the famed Ambiguous Case. The money can be in
only one other place. One more hole will do it.,}
And, as so many times before, Inspector Salty Miles was right....
* # *
If two sides and an opposite angle are known, one of the other
two angles may be determined by the law of sines and the third
from the sum of angles. Since sin 6 = sin(7i; - 0), a unique
solution is obtained only for a right triangle. When dealing
with other triangles, we find two triangles, a phenomenon
known as the ambiguous case.
Back to Inspector Miles: A triangle with two sides 15 and 10
units and an angle of 30° opposite the shorter side. Find the two
remaining angles.
Let the angle opposite the side measuring 15 units be a.
Apply the law of sines:
sin a sin 30°
15
10
15
sin a = —sin 30° & 0.75
a * 48.6°, or 131.4° .
The angle opposite the unknown side is, alternatively,
180° - (30° + 48.6°) = 101.4°
or
180° - (30° + 131.4°) = 18.6°
18.6°
131.4°
The angles of the triangle are either
30° 48.6° 101.4° ^
or
30°
131.4°
18.6°

482
Chapter 13 TRIGONOMETRY
Law of Sines for Spherical Triangles
For spherical triangles, that is, triangles whose sides are arcs
of great circles, we have
sin A sin B sin C
sin a
sin b
sin c
where A, B, and C are angles formed by tangent lines and
sides a, 6, and c are measured by their subtended central
angles.
To prove
sin A sin B
sin a sin b '
let ABC be any spherical triangle on a sphere with the center O.
From any point P on the radius OC, draw line PQ
perpendicular to plane OAB.
Through PQ and perpendicular to OA and OB pass planes
which meet radius OA and line OB in A' and B\ respectively,
forming angle PA 'Q = A and angle PB 'Q = B.
C
In the right triangle A PQ, we find
sin A' = sin A =
in the right triangle B'PQ,
siaB' = sinjB =
Z2-.
PAX '
in the right triangle OPB',
sin a =
in the right triangle OP A',
sin 6 =
PB1
OP ;
OP '
Consequently,
sin A PQ OP
sin a " PA' P5'
and we have
and
sin A
sin a
sin £ PQ OP
sin fc ~ PB' ' PA'
sin B
sin 6

Section 13.3 Solving Triangles
483
Analogously,
sin A
sin a
sinC
sin c
For a spherical triangle, the ambiguous case occurs in the
solution of an oblique triangle if given two sides and an angle
opposite one of them or given two angles and the side opposite
one of them.
Law of Cosines
VIETE Francois
(1540 -1603)
B
To solve a triangle when given
two sides and the included angle
or
three sides,
we use the law of cosines, first described by the French
mathematician Francois Viete:
a2 = b2 + c2 - 2 b c cos A
b2 = a2 + c2-2accosB
c2 = a2 + b2 - 2 a b cos C
To prove that
a2 = b 2 + c 2 - 2 b c cos A,
we apply the Pythagorean
theorem to the two right
triangles inside the triangle
ABC:
b2 = h2 + b2 cos2 A
and
a2 = h2 + (c-b cos A)2
Subtraction gives
b2-a2 = h2 + b2 cos2 A - [h2 + (c — b cos A)2]
a2 = 62 + c2-26c cos A .
The formulas
b2 = a2 + c2-2accos£
c2 = a2 + 62-2a6cosC
are proved in the same manner.
Find the length of the third side of a triangle with two sides
a and 6, and an angle a opposite side a.
Let the third side be x, and let the unknown angles be /3 and y\
draw an arbitrary triangle:
b/
b cos A
<
h
c -
c
V?
b cos A
—>
a

Chapter 13 TRIGONOMETRY
Apply the law of cosines:
x = V#2 + b2 - 2ab cos y
To find 7, first determine /3 by the law of sines:
sin /3 sin a
b ~~ a
b sin a
sin /3
a
We have the possibility that j3 is an acute angle such that
. (b sin a
p = arcsin
or
fi' =
n - arcsin
b sin a
a
The angle / is determined by the equation y = n - (a + /3);
since /3 is known, we find
7 =
6 sin a
7i - a - arcsin
a
or
7 =
b sin a
arcsin | I - a
a
By substituting for y in x = Va2 + 62 - 2a b cos /, we obtain the
third side,
v
a2 + b2 - 2 ab cos
7i - a - arcsin
b sin a
a
or
v°
2 + b2 - 2 a& cos
. (b sin a
arcsin | | - a
Find the length of the diagonals of a rhomboid with sides
35.0 cm and 46.2 cm and the acute angle 50°.
46.2 cm
Applying the law of cosines, we obtain
di = ^46.22 + 35.02 - 2 • 46.2 • 35.0 cos 130° * 73.7 cm,
d2 = ^46.22 + 35.02 - 2 • 46.2 • 35.0 cos 50° * 35.8 cm.

Section 13.3 Solving Triangles 485
Law of Cosines for Spherical Triangles
Sides of a Spherical Triangle
For spherical triangles, we have
cos c = cos a cos b + sin a sin b cos C .
where sides a, 6, and c, being arcs of great circles, are
measured by their subtended central angles.
Why? Let ABC be any spherical triangle on a sphere with the
center O. Draw the radii of the sphere to vertices ABC of
the triangle. Draw tangents to sides a and b at C and let
the tangents meet the extensions of OA and OB at P and Q,
respectively.
Applying the rule of cosines for the plane triangles OPQ and
CPQ and subtracting, we find
(PQ)2 = (OQ)2 + (OP)2 - 2 (OQ) (OP) cos c
(PQ)2 = (CQ)2 + (CP)2 - 2 (CQ) (CP) cos C
0 = (OQ)2 - (CQ)2 + (OP)2 - (CP)2
+ 2 (CQ) (CP) cos C - 2 (OQ) (OP) cos c,
and rearranging,
2 (OQ) (OP) cos c = [(OQ)2 - (CQ)2 ] + [(OP)2 - (CP)2]
+ 2 (CQ) (CP) cos C.
By the Pythagorean theorem,
(OQ)2-(CQ)2 = (OC)2 and (OP)2-(CP)2 = (OC)2,
and, consequently,
2 (OQ) (OP) cos c = 2(OC)2 + 2 (CQ) (CP) cos C.
Dividing through by 2 (OQ) (OP),
OC OC CQ CP
cosc=OQ'OP+OQ"OPCOsC
or
cos c = cos a cos b + sin a sin b cos C .
Cyclic permutation gives
cos a - cos b cos c + sin b sin c cos A
cos b - cos c cos a + sin c sin a cos B .

486
Chapter 13 TRIGONOMETRY
Angles of a Spherical Triangle
The law of cosines concerning angles of a spherical triangle
is here quoted without proof:
cos A = - cos B cos C + sin B sin C cos a
cos B = - cos C cos A + sin C sin A cos 6
cos C = - cos A cos 5 + sin A sin 5 cos c
Law of Tangents
FINCKE Thomas
(1561-1656)
If we know
two sides of a triangle and the included angle,
then, to find the remaining angles,
first use the law of cosines for the third side and
then - when all three sides are known - find the angles by
the law of sines or the law of cosines.
For a direct solution of the problem, we use the law of tangents,
first described in 1583 in Geometriae rotundi by the Danish
mathematician and physician Thomas Fincke:
a
a + b
tan^ (A -B)
tan J (A + B)
To prove the law of tangents, consider a triangle ABC where
angle A is greater than angle B. The external angle at C is
bisected by DE, which forms one side of a rectangle BDEF
about the triangle:
E ° cos a Nx y ' A a CQS &
b sin a
a sin a
c cos p
If half the external angle of C is a, then alternate angle CBF is
equal to a.
Let angle ABF be p.
In the above figure, the three right-angled triangles yield
CD = a cos a BD = a sin a CE = 6 cos a
AE = b sin a AF = c sin /? 5F = c cos /?.

Section 13.3 Solving Triangles
487
Opposite sides of a rectangle are equal,
a sin a = b sin a + c sin /3
c cos p = a cos a + b cos a
and, after rearrangement,
(a - b) sin a = c sin /3
(a + 6) cos a = c cos /3
Division gives
(a - b) sin a sin/3
(a + 6) cos a cos /3
a - b
a + b
tan a = tan /3
a - b tan /3
a + 6 tan a
The external angle at C equals the sum of the two opposite
interior angles of the triangle; therefore,
2a = A + B ; a= \ (A + B).
Since /3 = a - B, and a = -r (A+B), we have
/3 = \(A + B)-B
and finally
/3 = 7 (A-£),
a_h tang (A-B)
tan 2 (A +5)
Find the remaining angles of a triangle with two sides 9 and 6
and an included angle of 43°.
Use the notations of the triangle to
the right and apply the law of
tangents:
9-6
9 + 6
1
5
tang (A -B)
tan -(A+B)
A + B = 180° - 43° = 137° ; t: (A + B) = 68.5°.
1
5
tang (A-B)
tang (A -B)
tan 68.5°
tan 68.5°
= 0.5077... ; -r (A-B) * 26.9°

488
Chapter 13 TRIGONOMETRY
We solve the system of equations
\ (A + E) = 68.5°
\ (A -B) = 26.92°
A = 95.4°
B = 41.6°.
The sought angles are 95.4° and 41.6°.
The angles of a triangle are 50°, 60°, and 70°; the sum of its two
longer sides is 18 length units.
Find the lengths of the sides.
Let the longer sides be x and.y, and let the third side be z.
By the law of tangents,
70° - 60°
tan
x - y 2
18
70° + 60°
tan
x-y = 18
2
tan 5°
tan 65° '
x-y
x+y
= 18-
= 18
tan 5°
tan 65°
9tan5°
X " 9 + tan 65° * 9'37 ;
y * 18 - 9.37 = 8.63
By the law of sines,
9.37
sin 50° sin 70°
z & 7.64 cm.
The sides of the triangle are
9.37, 8.63, and 7.64 .

Section 13.3 Solving Triangles
489
Area of a Triangle
For a triangle ABC of sides a, 6, and c, we have
a b sin C a c sin B b c sin A
Area 2 = 2 = 2
This theorem is a corollary to the theorem stating that the area
of a triangle is half the product of the base and the height. In
the triangles ABC and DEF below, the height equals the product
of the sine of an angle (e.g., A and E, respectively) and the
length of the hypotenuse of a triangle whose side opposite the
employed angle is the height:
D f
hc = b sinA
b c sin A
AreaA5C =
hf = d sin E
AreaDEF =
dfsinE
The area theorem applies to both acute and obtuse angles.
Therefore, when the area and two sides are known, there are
two different triangles that satisfy the given conditions.
Find the angle C in a triangle ABC of area 24 cm2, where AC is
16 cm and BC is 6 cm.
16 • 6 sin C ^A . „ 1 „ rt^0 „,-
2 = 24 > sin C = -; C = 30° or 150° .
Two triangles are possible:
16 cm
16 cm

490
Chapter 13 TRIGONOMETRY
Surveying and Navigation
de JODE (or JUDAEIS) Cornells
(c. 1568-c. 1600)
Dutch geographer and
copperplate engraver
7>E QVADRANTE GEOMETRICO
IN QVO OVIDOVID
AD LINEARVM ET SVPERFICIERVM
VTPOTE XLTITVDINVM ET LATITVDINVM, Dl.
mention** facit lucutifsiaie tfcmoaftutur.
Sufnptibus&cjtpcnrisCefcNiiM di Iv&4itcdituf+
TCO R IE E R G AE,
TYPIS CHfclSTOPHOKl LOCHNER!.
<JW1 D. X C 1 I I L
Greek, geo, "Earth";
daiesthai, "to divide"
Latin, navis, "ship;
-igare (from agere), "to drive"
Geodesy is a branch of geology applying mathematics to
determine the shape and size of the Earth and its varying
magnetism and gravity.
Surveying is the science of making accurate measurements of
the Earth's surfaces. Land surveying includes both plane
surveying, which deals with areas sufficiently small to allow
the surveyor to disregard of the Earth's curvature, and geodetic
surveying, which takes into account the curvature of the
Earth's surface. Hydrographic surveying deals with contours
of the Earth under bodies of water.
Although the literal meaning of navigation is sailing, the
term refers to the art or science of setting a course at sea, on
land, or in the atmosphere.

Section 13.3 Solving Triangles
491
Locating Points and Measuring Distances
Ellipsoid, Spheroid: p. 405
Geo id
Greek, geoeides, "earthlike"
Parallels
Meridians
Greenwich Meridian
Longitude
Latitude
The Earth has an irregular, roughly ellipsoidal shape, called
a spheroid and generally referred to as a geoid. For the
problems in this text, however, we shall regard the Earth as a
sphere.
The poles are on the axis of rotation of the Earth. The plane of
the equator is imagined as passing through the center of the
Earth, perpendicular to the Earth's axis. Thus, all points on
the equator are equidistant from the poles.
Parallels (of latitude) are curves parallel to the equator;
meridians (of longitude) are curves perpendicular to the
equator. The parallels and meridians form circles of 360° in
circumference around the Earth. Meridians are all great
circles, whereas, among parallels, only the equator is a great
circle.
By international agreement in 1864, the 0° meridian passes
through the former London observatory at Greenwich,
England, and is generally known as the Greenwich meridian.
The longitude of a point is the angle between the plane of its
meridian and the plane of the Greenwich meridian.
Longitude is measured from 0° to 180° East and West of Greenwich.
The latitude of a point is the smallest angle formed between the
radius from the point to the Earth's center and the plane of the
equator. Latitude is measured from 0° to 90° North and South
of the equator. 0° is the equator itself; 90° North and 90° South
are the poles.
E
Geodesic
Bearing
In the figure, NS is the axis of the Earth; point A is the
intersection of the Greenwich meridian and the equator;
NPESEiPi is a meridian through a point P on the surface of the
Earth; EEi is the diameter of the equator; PP\ is the diameter
of the parallel through P. Angle POE is the^latitude of point P,
and AOE is the longitude of point P.
A geodesic (or geodetic) curve describes the shortest distance
between two points on a spherical surface, and is always an
arc of a great circle.
Bearings, or courses, are angles made with the meridians.

Chapter 13 TRIGONOMETRY
Two points A and B are located on latitude 40° North,
longitudes 10° West and 20° East, respectively.
Find the surface distance between A and B, along the North 40°
parallel, if the circumference of the Earth at the equator is
approximately 40 000 km.
Let the radius of the Earth be R and the radius of the parallel of
latitude 40° be r:
0 Meridian
S
We have
where
Thus,
An 10 + 20 rt
arcAB = 3&) -2nr,
r = R cos 40°
30 40000
AB = -^ • 2ti- cos40° * 2553.5 * 2550.
360
The sought distance is
2tc
2.55 • 103 km.
We have determined the surface distance along the 40° North
parallel, which is not on the arc of a great circle and, therefore,
does not mean the shortest surface distance between A and B.
Calgary, Canada, is located on latitude 51.03° North, longitude
114.05° West, and Gatwick Airport, London, U.K. is on
latitude 51.09° North, longitude 0.21° West, that is, nearly on the
same latitude.
The Earth's circumference is approximately 40 000 km.
Find the distance between Calgary and Gatwick
1. along the 51st parallel;
2. along the geodesic line connecting the cities.
--5^¾^

Section 13.3 Solving Triangles
493
Calgary
Gatwick
The arc CG of the 51st parallel is
p. 485
where
114.05-0.21 /rt ,
arc CG = — (2 n r)
40000
r = R cos 51° ; R = — ; 2 n r = 40 000 • cos 51° .
2k
Thus,
arc CG * ' • 40000 cos 51° * 7960 * 8.0 103 km.
To find the length of the distance along the geodesic line, apply
the law of cosines concerning sides of a spherical triangle,
cos n = cos c cos g + sin c sin g cos N .
Central angle c = 90° -51.09° = 38.91°;
g = 90°-51.03° = 38.97°.
Included angleN = 114.05°-0.21° = 113.84°.
Thus,
cos n = cos 38.91° cos 38.97° + sin 38.91° sin 38.97° cos 113.84°
n = 63.556°
and the distance is
63.556 • 40 000
360
^7 061 * 7.1 • 103 km.
The distance between Calgary and Gatwick is
1. along the 51st parallel is 8.0 • 103 km;
2. along the geodesic line is 7.1 • 103 km.

494
Chapter 13 TRIGONOMETRY
Find the distance along the geodesic line between a point A
located on latitude 34.84° South, longitude 59.09° West, and a
pointB on latitude 34.82° North, longitude 139.31° East.
The Earth's circumference is approximately 40 000 km.
Central angle 6 = (90°+ 34.84°) = 124.84°;
a = 90°-34.82° = 55.18°.
Included angle c = 180° - 139.31° + (180°-59.09°) = 161.60°.
cose = cos 124.84° cos 55.18° + sin 124.84° sin 55.18° cos 161.60°
* -0.965 55.
c = 164.92°, so the distance is
164.92 • 40 000
360
* 18 320 * 1.8 • 104 km.
Working All the Angles
In order to determine width, height, and depth of a distant
object - and its distance - we have a choice of measuring
angles of sight, of elevation, and of depression.
Lines of Sight
Sighting Angle
Angle of Elevation
Angle of Depression
When observing an object AB from a point O, OA and OB are
lines of sight, and the angle AOB is the angle of sight, or
sighting angle.
If we observe a vertical object CD from a point P, PC and PD
are sighting lines; C is sighted at an angle of elevation, D at
an angle of depression.
In the following examples we disregard the curvature of the
Earth and the fact that light does not travel in a straight line
when passing through air of varying density.
A radio tower is observed from a point 20 m above ground
level; the top of the tower is at an elevation of 30.53°, the base at
an angle of depression of 8.04°.
Find the height of the tower.

Section 13.3 Solving Triangles
495
Let the distance between the observation point and the tower
be x, the height h + 20 meters.
C
We have
elevation: h = x tan 30.53°
depression: 20 = x tan 8.04°
h_ _ tan 30.53 °
20 " tan 8.04°
tan 30.53°
h + 20 = 20— 5-^70-+ 20 * 103.5 m.
tan 8.04
A farmstead B is to be supplied with electricity by
underground cable from a distribution box A. To determine the
length of the cable, sightings are taken from two points on
the opposite bank of a river: point O situated on the
extension of line BA, and point P at a distance of 1530 m
from O, as shown in the drawing.
Sightings have given the angles
OPA = 32.3°
APB = 48.1°
POA = 82.8°.
Find the minimum length of cable required.

496
Chapter 13 TRIGONOMETRY
We have the following angles:
PAB
OAP
OPB
AOP + OPB
ABP
= 82.8°
= 180°
= 32.3°
= 82.8°
= 180°
Apply the law of sines to
triangle OAP:
, ^ sin
AP = 1530 .-^-
sin l
82.8°
S4.9°
+
—
+
+
—
32.3° = 115.1°
115.1° = 64.9°
48.1° = 80.4°
80.4° = 163.2°
163.2° = 16.8°
triangle BAP:
nA An sin 48.1°
BA = AP • . 1fiQO
sin 16.8
,„ort sin 82.8° sin 48.1° A^n
BA = 1530 ■ . . .- QO * 4317 m.
sin 64.9 • sin 16.8
A hot-air balloon is observed in the Serengeti due south, from
a point A at 22.23° elevation and, simultaneously, from a point
B 1500 m due south of A at 48.11° elevation.
Find the height of the hot-air balloon.
We have the "sighting angle" from the balloon,
48.11° - 22.23° = 25.88°,
and
- by definition:
h = y • sin 48.11°
- by the law of sines:
sin 22.23°
y = 1500 •
sin 25.88°
sin 22.23°. sin 48.11°
h = 1500 • . og QQO * 970 m.
sin 25.88
In an aircraft flying at constant speed v km/h and at an
altitude of 3200 m, the navigator observes an approaching
coastline at an angle of depression of -15.2° at time zero
seconds; 120 seconds later, an electronic circuit freezes the
inclinometer reading at -52.6°.
Find the speed of the aircraft.

Section 13.3 Solving Triangles
497
Let the distance between the two observation points be x m, and
the distance between the first observation point and the coastal
strip y m:
^^^^£^£^^
We have
and
x = 3200
v = 3200
sin 37.4°
x ~ y sin 127.4°
_ 3200
y " cos 74.8° '
sin 37.4°
sin 52.6° • cos 74.8° '
sin 37.4°
sin 37.4°
y ' sin 52.6°
3600
sin 52.6° • cos 74.8° 1000 120
* 280km/h.
A 20-m-high mast is placed on top of a cliff whose height above
sea level is unknown. An observer at sea sees the top of the
mast at an elevation of 46° 42', the foot at 38° 23' .
Find the height of the cliff.
h m
46° 42'
a = 90° - 46° 42' = 43° 18'; /3 = 46°42' - 38°23' = 8°19'.
We have
. ««o ^«. ™ sin a ^ sin 43° 18'
h = y sin 38° 23'; y = 20 • ,._ 0 = 20 • _._ QOinl
sin p
h = 20-S^l\8' >sin 38° 23'
sin8°19' '
= 20-
sin 8°19'
sin 43.30° • sin38.38c
sin 8. 32°
?* 59 m.

498
Chapter 13 TRIGONOMETRY
Alternative solution:
Let z be the distance from the observer to the plumb line through
the cliff-top pole.
h m
° AC\*
46 "42
We have
/i + 20 = z tan 46° 42'
-h = -ztan38°23'
20 = z (tan 46° 42' - tan 38° 23')
20
z =
h =
tan 46° 42'
- tan 38°
20 tan 38° 23'
tan 46° 42'
- tan 38°
20 tan 38.38°
23' '
23'
tan 46.70° - tan 38.38c
?* 59 m.
Determine the free space between a ski-lift chair and a
terraced area near the top of the hill. At the foot of a 35° incline,
the elevation to a lift chair directly over the terraced area
measures 46°; 60 m up the incline, it is 53°.

Section 13.3 Solving Triangles
499
a= 53°-46° = 7°. y =
p = 46°-35° = 11°. 8 =
By the law of sines,
180° -(a +p) = 162°.
180° - (90° - 46°) - p = 125°
x
60
sin / sin a
Also,
x
h =
x sin p
sin 8
sin 162° • sin 11°
sin p sin 8
= 60-
x =
60 sin /
sin a
sin 7° • sin 125°
_ sin 18° • sin 11°
= 60 • • „0 :—T7^~ * 35.4 m.
sin 7 • sin 55
Bearings and Cardinal Points
The bearing of, or direction to, a point on land or at sea may
be stated as the angle that the line of sight to the point forms
with the cardinal points of the compass - North, East, South,
and West.
There are two ways of stating a bearing.
The points Pi P<i P% P4 in the chart below have the following
bearings relative to the observation point O:
Px North 62° East or N62E
P2 North 20° West N 20 W
P3 South 18° West S 18 W
P4 South 46° East S 46 E
N20W
p2W
H.
N62°E
£
S46 E
The other method of stating a bearing, or course, is by giving
the clockwise (CW) angle from true North:
Pi bearing 62
P2 bearing 340
P3 bearing 198
P4 bearing 134

500
Chapter 13 TRIGONOMETRY
A ship at A sights a lighthouse L on bearing N65E and, after
sailing 8 nautical miles on course S45E to position B, the
same lighthouse at N32E.
Determine the distance between the ship when at B and the
lighthouse. (1 n.mi. = 1.852 km.)
► £
a = 90° - 65° = 25°
Angles BAL = 45°+ 25° = 70°
ABL = 45° + 32° = 77°
ALB = 180° - (70° + 77°) = 33
According to the law of sines,
BL 8
sin 70° sin 33
O j
BL & 13.8 n.mi. & 25.6 km.
A ship heading due north at a logged speed of 14 knots (kt)
sights at position A a lighthouse L at N20E and, at the same
time, a known shipwreck W at N45E; 30 minutes later, the
lighthouse is due east of the ship's new position B and the
shipwreck is at N 68 E.
Find the distance LW between the lighthouse and the
shipwreck. (1 n.mi. = 1852 m; 1 kt = 1 n.mi./h.)
The distance traveled by the ship is 7 n.mi.
We have, by the law of sines,
BW 7
sin 45° "" sin AWB '
where the angle AWB = 68°-45° = 23°; thus,
sin 45°
BW = 7-
sin 23*

Section 13.3 Solving Triangles
501
Observing that
and
BL = 7tan 20°
LBW = 90°-68° = 22°,
and applying the law of cosines to triangle BLW,
LW2 = BL2 + BW2 - 2 BL BW cos 22°
gives
LW
W
„ _0 sin2 45° rt tan 20° • sin 45° • cos 22°
tan2 20° + ■ 2
sin2 23°
?* 10.3 n.mi. & 19.1 km.
sin 23°
A ship S sights two navigation beacons, A at N 51.2 W and B at
N 32.0 E, situated 32 km apart on a line bearing N 78 E. To
steer clear of a reef off the spit of land at A, a course must be set
to take the ship past A at a distance of not less than 6.4 km.
Set the ship's course.
sin a
The sought course is N (51.2° + a) W.
We have
6.4
SA '
Noting that
p = 180° -32.0° -(180° - 78°) = 46.0°,
we can determine SA by the law of sines applied to triangle
ABSy
SA 32 32
sin 46.0° " sin (51.2° + 32.0°) " sin 83.2°
32 • sin 46.0°
SA =
sin 83.2°
We have
sin a =
6.4 6.4 sin 83.2°
SA
and
32 sin 46.0
51.2° + a = 67.2°.
- * 0.276; a & 16.0°,
The ship's course should be south of N 67.2 W.

502
Chapter 13 TRIGONOMETRY
13.4 Graphs, Domains, Ranges
Sine Curve
0
3rc/2 2tc
A point-by-point construction of the curves of the fundamental
trigonometric functions can be made from the unit circle;
below, that method is used to construct the sine curve:
y sin e
► e
Since the circumference of a circle equals the radius
multiplied by 2 k, and the radius of the unit circle is 1, the
circumference of the circle is 2 n, and the length of the curve -
representing one period - is 2 k radians.
Cosecant Curve
esc 6 =
sin 6
The cosecant is the reciprocal of the sine of an angle, and we
obtain the following graph:
9
n 3rc/2 2tc 571/2

Section 13.4 Graphs, Domains, Ranges
503
Greek, asymptotos,
"not falling together"
An asymptote is that straight line to which a curve moves
closer and closer.
As shown in the above graph, the asymptotes of the cosecant
curve coincide with the points where sine of 6 is 0. At these
points we have 1/0, which is undefined.
As with the sine, the length of one period of the cosecant is
2% radians.
Cosine Curve
*»o
The length of one period of the cosine function is 2% radians.
Secant Curve
sec 6 =
cos 6
The asymptotes of the secant curve coincide with the points
where cosine of 6 is 0:
**0
n Zn/2 2n 5rc/2
The length of one period of the secant function is 2% radians.

504
Chapter 13 TRIGONOMETRY
Tangent Curve
► 0
n Sn/2 2% 5n/2
The tangent curve approaches the vertical line = n/2
asymptotically, and this is repeated in periods of n radians; we note
that this is only half the length of one period of the sine, cosine,
secant, and cosecant functions.
Cotangent Curve
cot0 =
tan0
As a consequence of the cotangent being the reciprocal of the
tangent of an angle, the following graph is generated:
► 0
-n/2
n/2
n 3n/2 2% 5n/2
The length of one period of the cotangent function is n radians.

Section 13.4 Graphs, Domains, Ranges
505
Domains and Ranges
Trigonometric functions are periodic, which means that every
value of a trigonometric function corresponds to an infinite
number of arguments. To cover the entire interval [-1 , 1] of
sine and cosine, we can restrict the angle to the closed interval
K K~]
- 5" > 9* f°r sinx, and to the closed interval [0 , n] for cosx.
The corresponding inverse trigonometric functions are given
values in these intervals:
arcsm x:
arccos x:
K K
2' 2
[0,ii]
1-r
sinx
cosx
x
-3n/2 -n
3n/2
arccos x
x
arcsin x
tanx
cot x
Much the same argumentation applies for the functions tanx
and cotx and their corresponding inverse functions. Values
of tanx are restricted to the given open interval I - ;r , % I ,
cotx to the open interval ]0, n[; the corresponding inverse
trigonometric functions are
arctan x\
arccot x:
]— n r
"22 L
]0,k[.
x
-k/2
X
-n/2

506
Chapter 13 TRIGONOMETRY
There is no agreement regarding definitions of restricted
intervals for the functions secx and cscx and their inverses;
often used are
arcsec x:
arccsc x:
]-*-i].]°f]
The domain of the trigonometric function, restricted to the
given interval, is the range of its inverse, and vice versa, for
instance:
fix) A .
a domain:
tan x, ]-n/2, ti/2[
sin*
domain:
Hi/2,7i/2]
+~x
COS*
domain:
[0,tc]
tan*
i
range: [0,7t]
arccos x
range: [-n/2, n/2]
arcsin x
arctan*
x
range: ]-n/2, n/2[ ~%
arctan*
arcsin x

507
13.5 Trigonometric Identities
Trigonometric identities are equations that express relations
among trigonometric functions which are true for all values of
the variables involved. Such identities are of importance
when solving trigonometric problems and are powerful media
in calculus.
Fundamental Trigonometric Functions
Fundamental Identities
sin 6 =
cos 6 =
esc 6
1
sec 6
tan 6 =
cot 6 =
sin 6
cos 6
cos 6
sin 6
esc 6 = —
sec 6
sin 6
1
cos 6
Euler's Formula
EULER Leonhard
(1707-1783)
p. 792
The identity
elx = cosx + i sinx
where i is the imaginary unit and x a real number, is known
as Euler's formula, named after the Swiss mathematician
Leonhard Euler. In a later chapter, we shall show its validity
by using power series.
Since cos n = -1 and sin n = 0, we obtain
171 _
= "I,
by many regarded as the most beautiful formula in all of
mathematics.
Thus, the transcendental number e raised to the power of the
product of the imaginary unit and n, also a transcendental
number, yields exactly -1.
Similarly, since cos 2 n = 1, sin 2 n = 0, and e° = 1,
e2in = 1.
Cofunction Identities
The value of a fundamental trigonometric function of an acute
angle is numerically equal to the value of the cofunction of the
complementary angle.
K \
sin 6 = cos I — - 6
z J
K \
cos 6 = sin I — - 6
z J
K \
tan 6 = cot [ — - 6
K
cot 6 = tan I — - 6
J
\
J
K \
csc 6 = sec I — - 6
K
sec 6 = csc I — - 6
J
\
J

508
Chapter 13 TRIGONOMETRY
Even- Odd Identities
i
The cosine is an even function: cos 0 = cos( -0)
•- e
Being the inverse of the cosine, the secant is also an even
function.
The sine, cosecant, tangent, and cotangent, on the other hand,
are all odd functions: they have the same numerical values
for equal positive and negative angles, although with opposite
signs.
sin 0
+~ 0
The even- odd identities are:
sin(-0) = -sin 0 tan(-0) = -tan 0 csc(-0) = -esc 0
cos( -0) = cos 0 cot(-0) = -cot 0 sec( -0) = sec 6
Pythagorean Identities
^- X
Applying the Pythagorean theorem to a triangle in the unit
circle gives
sin2 6 + cos2 6=1.
Dividing both sides by cos2 6,
1
sin20 cos2 0
+ —
cos
20
cos
20
cos
20
and 1 + tan2 0 = sec2 0
Dividing by sin20,
sin20 cos2 0
+
sin20 sin20 sin20
and 1 +cot20 = csc20

Section 13.5 Trigonometric Identities
509
Compound-Angle Identities
To establish formulas for the sine, cosine, and tangent
functions of the sum (6 + <f>) and difference (6- <f>) of angles 6 and <j>
in terms of functions of the constituent angles, we refer to the
compound figure below.
We have
sin (6 + <S>) =
DE
AD
EF + FD BC + FD
AD
AD
_B(iAC_ FD_ CD_
" AD AC + AD CD
_B£_A£ FD CD
" AC AD + CD AD
sin (6+ <S>) = sin 0cos <f> + cos 6 sin <j>.
Replace <f> in (1) by (-<f>) to obtain
sin (6- ¢) = sin 0cos <f> - cos 6 sin <f>.
(1)
(2)
n
Replace 6 in (2) by [ — - 6 \ to obtain
K
sin I -- 6 - <f>
K
sin I — - 6 J cos <f>
K
- cos I — - 6 J sin 0
cos (6+ <f>) = cos 6 cos <f> - sin 6 sin <f>
Replace <f> in (3) by (-¢) to obtain
cos (6- <S>) = cos 6 cos <f> + sin 6 sin <f>.
(3)
(4)

510 Chapter 13 TRIGONOMETRY
To find a formula for tan (0 + 0), divide (1) by (3):
sin (0 + 0) sin 0 cos 0 + cos 0 sin 0
cos (0+0) ~ cos 0 cos 0 - sin 0 sin 0
sin 0 cos 0 cos 0 sin 0
1 + L
COS 0 COS 0 cos 0 cos 0
~ cos 0 cos 0 sin 0 sin 0
COS 0 COS 0 COS 0 COS 0
tan 0 + tan $
tan(0+^) = 1_tanetany (5)
Replace 0 in (5) by (-0) to obtain
tan 0- tan 0
tan 0-0) = -—- a . y . . (6)
r 1 + tan 0 tan 0
The cotangent is the inverse of the tangent function,
>m ., 1 1 - tan 0 tan 0
cotie+p; -tan(0+0 - tan 0 +tan 0
1 -
COt 0 COt 0
1 1
+
COt 0 COt 0
,/^ ,x COt 0 COt 0-1
cot 0+0)= — £-— . (7)
r COt 0 + COt 0
Replace 0 in (7) by (-0) to obtain
^/^ ^ cot 0 cot 0 + 1
cot (0-0) = — £-r-r" . (8)
r COt 0 - COt 0
Find the exact numerical value of sin 105°.
sin (60° + 45°) = sin 60° cos 45° + cos 60° sin 45°
_ ^1 (^l\ 1 (^ V6 + a/2
" 2 V27+2V2
Double-Angle and Multiple-Angle Identities
Formulas for the trigonometric functions of the double angle
and of multiple angles may be computed from the compound-
angle formulas.
Functions of 2 6
Equation (1),
sin (0+0) = sin 0 cos 0 + cos 0 sin 0 ,
gives, with 0=0,
sin 2 0 = 2 sin 0 cos 0. (9)

Section 13.5 Trigonometric Identities 511
Equation (3),
cos (0 + 0) = cos 0 cos 0 - sin 0 sin 0,
gives, with 0=0,
cos 2 0 = cos2 0- sin2 0 (10a)
= 2 cos2 0- 1 (10b)
= 1-2 sin2 0. (10c)
From Equation (5),
tan 0 + tan 0
tan (0+0) =
1 - tan 0 tan 0
we obtain
A _ 2 tan 0
tan 2 0
1 - tan2 0
and from (7),
0* (2*-l)|; 20* (2*-l)|; (11)
, , „ x COt 0 COt 0 - 1
cot (0+0 = — ^-rT">
r COt 0 + COt 0
cot2 0-1
cot20 = ——r-r- 0* n.7t; 20* n.n. (12)
Functions of 3 6, 4 0, and 5 6
sin 3 0 = sin (2 0+ 0)
= sin 2 0 cos 0 + cos 2 0 sin 0
= (2 sin 0 cos 0) cos 0+ (1 - 2 sin2 0) sin 0
= 2 sin 0 cos2 0 + sin 0-2 sin3 0
= 3 sin 0-4 sin3 0.
sin 4 0=4 sin 0 cos 0-8 sin3 0 cos 0.
sin 5 0 = 16 sin5 0 - 20 sin3 0 + 5 sin 0 .
cos 3 0 = 4 cos3 0- 3 cos 0.
cos 4 0 = 8 cos4 0- 8 cos2 0 + 1.
cos 5 0 = 16 cos5 0-20 cos3 0+5 cos 0 .
3 tan 0 - tan3 0
tan 3 0 =
tan 4 0 =
tan 5 0 =
cot 3 0 =
cot 4 0 =
cot 5 0 =
1-3 tan2 0
4 tan 0 - 4 tan3 0
1-6 tan2 0 + tan4 0 '
5 tan 0 - 10 tan3 0 + tan5 0
1-10 tan2 0+5 tan4 0
cot3 0 - 3 cot 0
3 cot2 0-1
cot4 0-6 cot2 0+1
4 cot3 0 - 4 cot 0
cot5 0-10 cot3 0 + 5 cot 0
5 cot4 0-10 cot2 0+1

512 Chapter 13 TRIGONOMETRY
Functions of the Half-Angle
Equation (10c) may be rearranged as
. „ _ 1- cos 20 . 9 0 1-cos 0
sin'2 0 = - ; sin'2 ^ = r>
m2 = ± Y~
. - ^i- cos 0 /Hr>x
sin - = ± A/ 2 (13)
and Equation (10b) as
o _ 1 + cos 20 0 0 1 + cos 0
cosz 6 = : cosz — = -
I-aP
^.-. cos 0
cos £ = ± \ 7T . (14)
Dividing (13) by (14), we have
6 ^ /l - cos 0 6 ^ /l + cos 0 __ ,^ „.
'"r^u^fl ; ^r^i-cosr (15)(16)
Conversion of Functions to the Half-Angle Tangent
To solve trigonometric equations, it is often useful, or
necessary, to have all trigonometric quantities expressed in
one function only, usually the tangent of the half-angle.
We have, from Equation (9),
sin 6 = 2 sin ~ cos ~ ,
where we introduce a denominator
O C/ o 6 -i
sinz — + cosz 9 = 1
to the right-hand side of the equation and divide both the nu-
merator and the new denominator by cos2 -z to obtain
2
•
sin
0
2
cos2
o 6/
sinz —
2
cos
0
2
+ cos
0
2
2£
2
2 6 6 o 0 0 0
sin — cos ~ cosz ~ 2 tan ~
sin0 = 2 n = n n~~ = n • (17)
sinz ~ + cosz — sinz — + cosz ~ 1 + tanz —
cosz —
We further have, from Equation (10a)
n 9 6 .96
cos 6 = cosz — - sinz — ,
which, by analogous treatment, is transformed into

Section 13.5 Trigonometric Identities
513
o P o 0
cosz — - sinz —
cosz ~- sinz — cosz ~ l-tanz —
cos 0 = - r = £— = . (18)
sinz — + cosz — sinz — + cosz — 1 + tanz —
cosz ~
The remaining formulas are:
2 tan ~ 1 + tan2 -
tan 0 = - esc 0 = ^- (19) (21)
1 - tan2 - 2 tan -
1 - tan2 - 1 + tan2 -
cot 0 = — sec 0 = - (20) (22)
2 tan — 1 - tan2 —
K
Since tan — is undefined, these conversion formulas are not
valid for angles 6 = n . n .
Converting Sums of Functions to Products
Add and subtract the compound-angle formulas (1) and (2), in
the form
(1): sin (p + q) = sinp cos q + cosp sing
(2): sin (p-q) = sinp cos q - cosp sing,
(1 + 2): sin (p + q) + sin (p-q) = 2 sinp cos q
(1-2): sin (p + q) - sin (p-q) = 2 cos p sin q
where we substitute
0+0 0-0
p + q = 0; p-q = ¢1 P =—— ; q = —£-
and obtain
sin 0+sin 0 = 2sin(—— j cos (—J^) (23)
sin 0- sin 0 = 2 cos f —T^J sin (^-^) • (24)
In an analogous manner, Equations (3) and (4) will produce
the equations
fe+ d\ (0- <b\ , x
cos 0 + cos 0 = 2 cos I —-— I cos I —-— I (25)
cos 0-cos 0 = -2 sin I—-^ J sin I—^1. (26)
We quote without proof:
tane+tan0=Sinle+^; cot0+cot0 = 4^^- (27)(29)
r cos 0 cos 0 sin 0 sin 0
tane-tan^=sin^-^; cote-cot0 = - "."^T^ (28)(30)
cos 0 cos 0 sin 0 sin 0

514 Chapter 13 TRIGONOMETRY
Breaking up Trigonometric Products
Formulas for breaking up products sin 6 cos 0 and cos 6 sin 0
may be obtained by adding and subtracting the compound-
angle formulas (1) and (2),
(1) sin 6 cos 0 + cos 6 sin 0 = sin (6 + 0)
(2) sin 6 cos 0- cos 6 sin 0 = sin (0- 0) .
Addition and division by 2 gives
(1 + 2) sin 0 cos 0 = J [sin (0 + 0) + sin (0 - 0)] . (31)
Subtraction and division by 2 gives
(1 - 2) cos 0sin 0 = ^ [sin (0+0)- sin (0 - 0)] . (32)
To break up products sin 0 sin 0 and cos 0 cos 0, we add and
subtract Equations (3) and (4),
(3) cos 0 cos 0 - sin 0 sin 0 = cos (0 + 0)
(4) cos 0 cos 0 + sin 0 sin 0 = cos (6- ¢).
Subtraction and division by 2 gives
(4 - 3) sin 0 sin 0 = -r [cos (0-0)- cos (0 + 0)] . (33)
Addition and division by 2 gives
1
2
(4 + 3) cos 0 cos 0 = o [cos (0-0) + cos (0 + 0)] . (34)
Inverse Trigonometric Functions
As a corollary of the trigonometric cofunction identities, we
have the inverse trigonometric identities, here quoted without
proof.
sin (arcsin x) = x ; arcsin (sin #)=# if - — < # < —
cos (arccos x) = # ; arccos (cos #)=# if 0 < jc < 71
pp. 505 - 6 e£c. intervals: pp. 505 - 6
arcsin (- x) = - arcsin x
arccos (- x) = n- arccos x
arctan (- x) = - arctan x
arccot (- x) = n - arccot x
% ) X v 1 — X
arcsin x = :r- arccos x = arccos Vl-*2 = arctan-7 = arccot x> 0
2 VT^2 *
7t 1 V 1 — # #
arccos # = — - arcsin x = arcsin V1 - *2 = arctan = arccot . x> 0
z # V1 - #2
71 X 11
arctanx = 5-- arccot# = arcsin . = arccos , = arccot — #> 0
z V1 + #2 Vl + *2 *
71 1 X , 1
arccot x = — - arctan # = arcsin , -= = arccos . = arctan — x> 0
z yl+x2 Vl + x2 *

Section 13.5 Trigonometric Identities
515
arctan —
x
x>0
arccot x = <
v^
arctan — + n x < 0
x
arcsin x =
arccos x =
arctan x =
- i • In (i x + Vl - x2)
- i • In (x + V x2 - 1)
1 1 + ix
2l lnl
arccot x = -
1 x
1 ix + 1
•r-r • In: 7
2 1 1 x - 1
1 i x - 1
77-r • In: 7-
2 1 ix + 1
ifx> 0
Sums and Differences
= arcsin
iin (x V 1 - y2 ± y ^fl-x2) x2 + y2 < 1 ; xy < 0
arcsin x ± arcsin y < = 71 — arcsin \x V 1 -y2 ± y yl-^x2) x>0; y > 0; x2 +y2>l
<= - n - arcsin (x V 1 -y2 ± y Vl - x2) x<0; y < 0 ; x2 +y2>l
arccos x + arccos y
arccos x - arccos y
\-
arccos [xy - V(l - x2) (1 -y2)J
arccos [xy + V(l - *2) (1 -y2)J
- arccos [xy + V(l - *2) (1 -y2)J
= arctan
* + y
1 -xy
arctan x + arctan y </ = ^ + arctan
x +y
= -71: + arctan
1 -xy
x + y
v<
1 -xy
(x+y)>0
x <y
x >y
xy < 1
x > 0 ; xy > 1
x < 0 ; xy > 1
= arctan
x - y
1 + xy
arctan x - arctan y J = ^ + arctan
x -y
1 + xy
= -71 + arctan
x - y
1 + xy
xy > - 1
x > 0 ; xy < - 1
x < 0 ; xy < - 1
arccot x + arccot y = arccot
arccot x - arccot y = arccot
xy- 1
x +y
xy + 1
y - x

516 Chapter 13 TRIGONOMETRY
13.6 Trigonometric Equations
In trigonometric equations, the unknown quantity appears in
the form of trigonometric functions of one or more variables.
If only one kind of trigonometric function is present, the
equation is said to be basic and may be solved in the same
manner as algebraic equations. If the equation contains
several functions of one angle or several angles, it must be
reduced to basic form.
Because of the periodicity of trigonometric functions,
trigonometric equations have an infinite number of solutions unless
restricted by a side condition; all solutions must satisfy both
the equation and the side condition.
Positive solutions less than 2% (360°) should always be
determined first; all other solutions will follow automatically.
Tf f arcsin a 1
If sin x = a * = 1 r + n • 2 ft
I n - arcsin a J
esc x = a
{arccsc a 1
n - arccsc a 3
cos x = a x = ± arccos a + n • 2 n
sec x = a x =± arcseca + n • 2 n
tan x = a x = arctan a+ n • n
cotx = a x - arccot a + n >n
where n is an integer.
In a pure trigonometric equation, the unknowns appear only
in the form of trigonometric functions; in a mixed
trigonometric equation, they will be found also in algebraic form.
Mixed trigonometric equations can generally be solved only
by graphical methods or by numerical analysis.
What is said here about trigonometric equations applies also
for trigonometric inequalities.
Basic Trigonometric Equations
Solve the equation
sin x = -r ,
in the general case and with the side condition 0 < x < n
I arcsin - I g-
X =\ ,^ + 71.271=^,-^ + 71.271,
1 5 71
n - arcsin —
v<
6 J
where n is an integer.

Section 13.6 Trigonometric Equations 517
Of these, the solutions
*1=|. = 30° ; x2 =^= 150°
satisfy the side condition.
Solve the equation tan x= —y2 .
An electronic calculator gives
x * - 0.955 + 7i. 7i * - 54.717° + n . 180°
?* 125.28° + n • 180°, where n is an integer.
cot2 x
Solve the equation —-— = 1.
cot x = ± 2 .
Since the arc cotangent function is usually not available on
calculators, we use its reciprocal,
x = arctan I ± — I + n • n & ± 0.464 + /1-71:,
where n is an integer.
Solve the equation cos 3 x = -r-; 0 < x < 2 n .
K
Sx= ± 7T + n-2n
o
n 2x 12 n ± 1
X=±18 + n-"F=_l8— •*•
_ JL ll7C 137t ^% 257t 357t
X " 18 ; 18 ; 18 ; 18 ; 18 ; 18
Reducing the Trigonometric Equation
The simplification of trigonometric equations involving
several functions of several angles to basic form is an
integral part of the task of solving trigonometric equations.
To effect this transformation, we have recourse to a number
of trigonometric conversion formulas and Pythagorean
identities which are helpful when handling trigonometric
terms that include multiples of the unknown variable; the
decision of which identity to choose generally depends on the
form of the equation.
The introduction of a new variable is sometimes beneficial
in order to make calculations simpler, although this method
should be used with discretion. The purpose is often to bring
the equation into a factorable form and, subsequently, to equate
the individual factors to zero.

518 Chapter 13 TRIGONOMETRY
To solve a trigonometric equation that contains different
trigonometric functions, one must arrange the equation so
that it contains terms of one and the same trigonometric
function (exclusively sine, exclusively cosine, exclusively
tangent, etc.).
Converting all trigonometric functions present into tangent
functions of the half-angle is often a profitable form of
substitution, provided that it does not lead to equations higher
than the second degree; it must be kept in mind, however, that
this method cannot be employed for an angle k, as tan tt is
undefined.
There is no general rule, however, that will unconditionally
provide the solution of a trigonometric equation; ingenuity
and perseverance are often the prerequisites of success.
If algebraic methods fail, we might fall back on graphical
solution or numerical analysis.
Employing Double-Angle Identities
• Solve the equation
sin x cos x = —; 0 < x < -r-.
4 2
We have
p. 510, Eq. (9) 2 sin x • cos x = sin 2 x = — ,
2
which gives
k
a
6
2* =S5k
6 J
> + n-2n; x = "\ *- f + n -n .
5 k 3 0k
In the interval 0 < x < -r— = 19 , we have the following
solutions:
n 5k
Xl = 12 X2 = 12
_ 13n 17 k
Xs ~ 12 *4 " 12
_ 25 k 29 k
*5 " 12 *6 ~ 12
Converting to Half-Angle Tangent
Equations that contain different trigonometric functions
of one and the same angle may be rendered into a form
containing exclusively tangent functions of the half-angle.
This method is advantageous except when it leads to equations
higher than the second degree.
It is important to observe that the method is not valid for an
angle k rad; and since tan -r does not exist when half-angle
tangent identities are employed, one must test the equation for
the validity of a possible solution = k .

Section 13.6 Trigonometric Equations
519
Solve the equation
<3~
cot x - csc x - -~ =0; 0 < x < 2 ti; .
p. 512
By substituting half-angle tangents, we have
1- tan2— 1+ tan2 —
2tan-
and, after reduction,
x
2tan-
V3
3
1- tan2f - 1-tan2! =2--^-
tan2— + — • tan— = 0
tanf
x ( x V?
tan J tan- + -
0.
X
tan — = 0
x
— = 0 + 71-71
x = 0 + n- 2% .
There is no solution < 2 n ;
rejected.
tanf
X
2
Vs
x =
5ti;
6
5k
3
x = n cannot be a solution since cot n and csc n are undefined.
5 7C
Testing for x = -r- gives
5 k 5k ^
cot—-csc—- —
V5
3 +^
V3
3
0.
In the interval 0 < x < 2 rc, the equation has only the solution
5k
3 '
x
Factoring
Like algebraic equations, trigonometric equations can often
be solved by factoring with the help of trigonometric identities.
Which identity should be chosen depends entirely on the type of
equation.
Employing Pythagorean Identities
A Pythagorean identity may be conveniently employed to
convert terms into one and the same trigonometric function
and render the equation suitable for factoring. Conversion
into tangent functions of the half-angle would, in this case,
produce a fourth-degree equation.

520
Chapter 13 TRIGONOMETRY
Solve the equation
2 cos2 x - sin # = 1; 0<x <2n .
Write
2 cos2 x - sin #-1 = 0
and obtain, successively,
2(1 - sin2 x)- sin x -1 = 0
2 sin2 x + sin #-1 = 0
(2 sin #-1) (sin #+1) = 0 .
Equate each factor with 0 :
2 sin #-1 =
sin# =
n 5k
X = 6 ; 6
0
1
2'
•
sin# + 1
sin#
_ IE
= 0
= -1
The solutions are
K
5n
3k
*1 — f* > %2— ^ > *3 ~~ o
Converting Sums of Functions to Products
When every trigonometric term features a different multiple
of the variable, it is sometimes possible to accomplish
factoring by converting sums of trigonometric functions to products.
p. 513
Solve the equation
sin 8 # - cos 5 # = sin 2 #.
Rearrange,
(sin 8 # - sin 2 # ) - cos 5 # = 0 ;
and transform (sin8#-sin2#) into a product,
2 cos
8# + 2 #
8 # - 2 # ,
sin | I - cos 5 # = 0 ;
2 cos 5 # sin 3 # - cos 5 # = 0 ;
cos 5 # (2 sin 3 #- 1) = 0 .
Equate each factor with 0 :
cos5# = 0
K
5# = ± — + n . 2k
k
n • 2 k
sin3# = —
3# = — + n • 2k
d
K
71 . 2 71
The infinitude of solutions is
K
Xl = ±w +
n • 2 k
; *2 = if +
K
18
n >2k
.2k
; *3
•3 =
5k
; 6
5 K
= 18
5k
+
+
n
n •
71
.2
3
27t;
.2rc
3
K
where n is an integer.

Section 13.6 Trigonometric Equations
521
p. 511
Conversion to Functions of the Same Angle
Trigonometric identities are employed to produce functions of
equal angles.
Solve the equation
4 cot 4 x
= 1; 0<x<2ti;
1 - 3 cot 2 x
and give the answer to three reliable digits.
The equation contains only one trigonometric function, but of
two multiples of the unknown.
We consider 2x as the unknown variable, and write
4cot2(2x) = l-3cot2x,
where we insert
cot 2 (2 x) =
cot2 2x-l
2 cot 2 x
and obtain
cot2 2 x - 1
= l-3cot2x
2 cot 2 x
4 cot2 2x-4 = 2 cot 2x - 6 cot2 2x.
By rearranging and dividing by 2,
5cot22x -cot2x-2 = 0,
we have a second-degree equation in cot 2x with the solution
i±Vii
cot2jc =
10
2 x = arccot —rr— + n • n.
Most calculators have no key for the arccot function, and we
write instead
10
2x = arctan
1± V5
+ ft • K.
In the interval 0<>x <2n,
10
# = o" f arctan
1 + V41
+ 71 • 71
* * 0.4668 + 7i • r-.
*1 * 0.4668 + 0 = 0.4668
x2 * 0.4668 + 1.5708 = 2.0376
x3* 0.4668 + 3.1416 = 3.6084
x4 * 0.4668 + 4.7124 = 5.1792
X = 2 \ arc^an
10
1- V4L
K
+ n • n
x **- 0.5377 +1-17.
*5 * - 0.5377 + 1.5708 = 1.0331
x6 * -0.5377 + 3.1416 = 2.6039
x7 * -0.5377 + 4.7124 = 4.1747
x8 * - 0.5377 + 6.2832 = 5.7454
The eight solutions of the equation in the interval 0 < x < 2 n
are:
x\ = 0.467 x5 = 1.03
x2 = 2.04 x6 = 2.60
#3 = 3.61 #7 = 4.17
#4 = 5.18 xg = 5.75

522 Chapter 13 TRIGONOMETRY
Introduction of a New Variable
Solve the equation
2(1-sina) = cos* ; 0 < x < 90°
Rewrite,
sin x + 0.5 cos x = 1 .
Substitute
0.5 = tan u ; u & 26.57° .
sin u
sin x + cos x = 1
cos u
sin x cos u + cos # sin u — cos a
sin (x + u) = cos u & 0.8944.
x + u * 63.43°; 116.57°
xi = 36.87° ; x2 = 90° .
Alternative solution:
2 (1 - sin x) = cos x
4-8 sin x + 4 sin2 # = 1 - sin2 #
5 sin2 x - 8 sin # + 3 = 0
sin # = — ± —
o o
xi = 36.87° ; x2 = 90° .
Solve the equation
2 cos #-sin # = -2; 0<#<15.
Rewrite,
sin x - 2 cos # = 2.
Substitute
tan u = 2 ; ^ ** 1.1071 .
sin u
sin # - cos x = 2
cos u
sin # cos a - cos x sin ^ = 2 cos u
sin (# - u) - 2 cos ^ = -=■
V5
#- a ?*
/1.1071 1
lrc-1.1071 J +n'2n
f 2.2142 1
X * |K J + 71 • 2 7C,
where /z is an integer.
In the interval 0 < x < 15, we have:
xi = 2.21...+ 0-271 * 2.21
#2 = n ••• + 0 • 2 7i = 71
x3 = 2.21...+ 27i * 8.50
#4 = 71...+ 2ti = 3ti ** 9.42
x5 = 2.21...+ 4k * 14.78

Section 13.6 Trigonometric Equations
523
Alternative solution:
After substituting the half-angle identities, we have
/l-tan2|X
V
X
1 + tan2 rr
2 tan 2
2/
V
X
1 + tan2 — .
+ 2 = 0
and, after reduction,
x
tan— = 2.
|-* 1.1071 + n.n
x & 2.2142 + n • 2 7C, where n is an integer.
Testing the original equation for x = n (where tan x is
undefined), we have
2 cos n - sin n = 2 (-1) - 0 = - 2,
which shows that
x = K + n -2k = (2n + l)n
is also a solution.
In the interval 0 < x < 15, we have:
X\ =
X2 =
*3 =
X4 =
X5 =
2.21...+ Ok
(2 • 0 + 1) K = K
2.21...+ 2k
(2 . 1 + 1) 7C = 3 7C
2.21...+ 4k
* 2.21
* 3.14
* 8.50
* 9.42
* 14.78
Extraneous Solutions
Extraneous solutions may occur when both sides of an
equation are raised to a power or when both sides are multiplied by
an expression that contains the variable. Solutions obtained
after such manipulations must be verified by substitution into
the original equation.
Solve the equation
sin x = Vl — cos x; 0 < x < 2 n .
Square both sides of the equation,
sin2 x = 1 - cos x
sin2 x - 1 + cos x = 0 ;
Pythagorean Identities: p. 508 we obtain
- cos2 x + cos x = 0
cosx (1 -cosx) = 0
Equate each factor with 0 :
cosx = 0
K
x = ± — + n n .
cosx = 1
x = 0 + n.2k .

524
Chapter 13 TRIGONOMETRY
Pythagorean Identities: p. 508
In the interval 0 < x < 2 n, we have the potential solutions
n 3_rc
2
*1 =2;X2
X3 = 0
To verify the solutions, insert x\> x^, and x% in the original
equation:
inf-V
sin
sin
1-cos | = 1-1 = 0
r-V
3 71
cos -r- =-1-1 = -2^0
sin
0 - Vl -cos 0 = 0
3 71
The results show that X2 = ~^~ is an extraneous solution and
that the equation has only the solutions
—
•A* ™~ V/ * •& ™™ ^ •
Solve the equation
sec x - tan ac - 1 = 0 ; 0 <x < 2 tx .
tan ac + 1 = sec x
(tan x + 1)2 = sec2*
2 tan x = sec2 x - (tan2 x + 1) = sec2 x - sec2 x
tan x = 0 .
In the interval 0 < x < 2 n, we have the potential solutions
x1 = 0
Checking the results,
sec 0 - tan 0-1
= 1-0-1 = 0
*2
71 .
sec 7i - tan n - 1
= -1-0-1 = -2,
shows that 7t is an extraneous solution and that the equation
has only the solution x = 0
Alternative solution:
sin x
cos x cos x
= 1
1 - sin2 x
1-
1 - sin x = cos x
(1 - sin x)2 ■
2 sin x + sin2 x = 1 - sin2 x
sin x (sin x - 1) = 0
sinx = 0
x = 0.
sinx = 1
71
x =— ; rejected since secant and
^ 71
tangent are undefined at —

Section 13.6 Trigonometric Equations
525
Impossible Solutions
Solve the equation
2 cos2 x + 5 cos x = 3 .
cosx = —
2 cos2 x + 5 cos x - 3 = 0
(2 cosx-1) (cosx + 3) = 0
cos x = -3 ;
x = ± — + 2k n
where n is an integer.
rejected,
as | cos x | must be < 1
Solve the equation
2 sec x - cos x + 1 = 0 ; 0 < x < 2 n;.
- cos x + 1 = 0 .
cos x
Multiply both sides of the equation by cos x and rearrange,
2 - cos2 x + cos x = 0
cos2 x - cos x - 2 = 0
(cosx-2) (cosx + 1) = 0
cos x = 2 ;
rejected.
cos x = - 1
x = n .
Checking,
2 sec n - cos k + 1
=-2+1+1
= 0
The equation has only the solution
x = n .
Lost Solutions
When both sides of an equation are erroneously divided by the
same factor, one or more solutions might get lost.
Solve the equation
tan x cos2 x = sin2 x ; 0 < x < 2 n .
siii x cos2 x
in2x = 0
sin
cos x
sin x(cos x - sin x) = 0.
sinx = 0
x = 0 ; n .
cos x - sin x
tanx
x = —
0
1
4 ; 4

526
Chapter 13 TRIGONOMETRY
Alternative solution:
tanx =
sin2x
cos2 x
= tan2x
(which must not be reduced to tan x = 1),
tanx (1 -tanx) =
tan x = 0
x\ = 0
x2 = n
tan x
*3
x3 =
0.
1
k
4
4
Systems of Trigonometric Equations
p. 513
Solve the system of equations
sin x + sin y
x+y =
1
5k
~3~
Transform the left-hand side of the first equation into a
product,
x+y x — y
sin x + sin y = 2 sin —-— cos —-—,
J 2 2
where
Consequently,
o • x+y o • 5k i
2 sin —-— = 2 sui-tt- = 1.
Z b
x —y , x-y
cos —-— = 1 ; —-— = n-2n.
We have
which gives
x—y -n•4 k "
5k
x+y = -3-
5k ^^
x = -77- + n • 2k
5k
y =-t- - n-2n
where /z is an integer.
Solve the system of equations
sin x sin y = —
1 ^
2
x+y =
2k
j

Section 13.6 Trigonometric Equations
527
sin x sin | -r— - x ) = 9, which may be written
Pythagorean Identities: p. 508
2 n
2 K
sin x I sin -t— cos x - sin x cos —
and, finally,
1
2
V3
sin x 1 —t- cos x + — sin x I = —
sin x (V3 cos x + sin x) = 1
V3 sin x cosx - (1 - sin2 x) = 0
*v3 sin x cos x - cos2 x = 0
cos x (^/3 sin x - cos x) = 0.
Equating each factor with 0, we have
cosx = 0
K
x\2 - - W+n ' 2n .
yi
2%
K
-X\
= — -n • 2 n .
0
y2
2k
7k
x2
-ra • 2 n .
The solutions are
sin x - cos x
0
tanx =
K
X3 = — + n • 71.
y3
2 71
71;
*3
= 2 -n.ic
7C
Xi = — + 71 • 2k
K
y1 = - -n.2n
j
X2 = -^-+/1-271:
7ti;
y2 = -7- - n • 2 jc
0 >»
7C
X3 = — + n • n
n
y3 = 2-nKJ
where n is an integer.

528
Chapter 13 TRIGONOMETRY
13.7 Limits
The Inclusion Theorem
The inclusion theorem, intuitively obvious and here quoted
without proof, is often useful when evaluating complicated
limits.
If
and
then
lim f(x) = lim g(x) = A
x —> a x —> a
fix) < h(x) < g(x), for all x near a,
lim h(x) = A .
x —> a
Given
lim [--
JC-> oo ' X
lim —
JC-> oo X
= 0
show that
lim
X —> oo
sin x
x
= 0.
The value of a sine function varies between -1 and 1; thus, for
x > 0,
1 sin x 1
-- < <- .
•A/ *As *As
Since lim I - —
JC-> oo • X
lim — = 0 , we have
x
X —> oo
sins
lim = 0.
X —> oo
X
*

Section 13.7 Limits
529
Two Crucial Limits
We shall discuss two limits that are of special importance for
finding derivatives of trigonometric functions.
Isin x . _.
lm = 1 : x in radians.
Proof:
x->0
X
In a unit circle, with
O at the center,
OAB a sector of a unit circle,
C the intersection of an extension of line
segment OB and a tangent to the circle at A ,
Z AOC = x rad,
we have
OA = OB = 1,
AC = tan x ,
and
area of triangle AOB = —-— ,
x
X X
area of sector AOB = -— • k = —
2 k 2
area of triangle AOC = ~ (l)tanx
tanx
Comparing these areas, we find
sin x
2
=> 1
■ <
< ■
X
2
X
<
tan
2
■ < -
X
1
sinx < x <
sinx
cos x
sin x cos x
sinx
=> 1 > > cos x
x
As x tends to 0, cos x tends to 1 and since sine is an odd
function, with
sin (-x) -sinx sinx
we have for positive and negative values of x that
_. sinx
lim = 1 .
cos*
x->0
X
QED
-** -4k |V/
-2k
flx)= -¾
sin*
2k

530
Chapter 13 TRIGONOMETRY
1 - COS X .
lim = 0 ; x in radians.
Proof: Multiply the numerator and denominator by (1 + cosx):
Pythagorean Identities: p. 508
1-cosx ,. l-cos2x
lim = lim —7z r = lim
sin2x
x->0
X
„ _^ r, x (1 + cos x) ~ x (1 + cos x)
X —> U X —> U
= lim
sin x sin x
x
1 + cos x I
As sin 0 = 0, cos 0 = 1, and lim
sinx
x->0
X
= 1, we find
1 - cos x
lim = 1
0
1 + 1
= 0.
QED
Building New Limits
To evaluate
lim
tanx
x
use the values
sinx
lim =1; limcosx=l
x->0
and write
x
sin x
*->o
X
x->0
X
x->0
,. tanx ,. cos x
lim = lim = lim
x->0
sin x
x
cos x
= 1
To evaluate
lim
sin 5 x
2x
substitute y = 5 x, use the value lim
siny
= 1, and write
lim
sin 5 x
2x
= lim
siny
lim
sin y 5_\ 5
1 2) = 2
*

Section 13.7 Limits
531
Oscillating Values
Some oscillating functions fail to have a limit value when x
tends to 0. We shall discuss the behavior of the functions
fix) = sin (1/%) and fix) = cos il/x) as x tends to 0.
fix) = lim sin — does not exist.
*->0 x
Why? As
sin - = 1;
. 3 7C
sin — = -1;
sin x = sinO* + n • 2n) for any integer n.
we have
if
sin — =1
x
1 * c
— = — +2nn
x 2
or x =
ji(1 + 4/i)'
where n is an integer.
if
or
sin
1_
X
X —
X
3k
2
ni3
where n
-1
+ 2n n
2
+ 4n)'
is an integer
Thus, ifx tends to 0, then fix) oscillates between -1 and 1, and
no limit value exists for fix) = lim sin — .
x
By a similar line of reasoning, one can also show that
lim cos — oscillates between -1 and 1 and, thus, does not exist.
x
fix) = sin (1/jc)
fix) = cos (1/jc)

533
Chapter
14
HYPERBOLIC FUNCTIONS
Page
14.0 Introduction 534
14.1 Fundamental Hyperbolic Functions 536
14.2 Inverse Hyperbolic Functions 539
14.3 Identities 541

534
Chapter 14 HYPERBOLIC FUNCTIONS
There once zvas a tqiitterfrom Wates
Who wanted to increase her sates.
She tqdtfat and thin,
And she never gave in.
Sweaters hyperbolic sold in bates.
14.0 Introduction
LAMBERT Johann Heinrich
(1728 -1777)
The analogy between the equation x2 + y2 = 1, which represents
a circle whose radius is 1 and whose center is at the origin
of an orthogonal coordinate system, and that of a hyperbola,
x2 -y2 = 1, started to attract interest from mathematicians in
the latter part of the 17th century.
Since the equations of the circle and hyperbola differ only by a
minus sign, and their areas may be expressed in
trigonometric and logarithmic functions, respectively, the idea
developed that the imaginary unit was involved in a relation
between trigonometric functions and the logarithmic function,
the circle being represented by trigonometric functions and the
hyperbola by hyperbolic functions - that is, factors of the
imaginary unit and trigonometric functions.
Many participated in the development of hyperbolic functions,
but the most comprehensive early publications (1768, 1770)
were by the German mathematician J. H. Lambert, who
introduced the names and notations that we still use.
sinus hyperbolicus
cosinus hyperbolicus
tangens hyperbolicus
cotangens hyperbolicus
secans hyperbolicus
cosecans hyperbolicus
sinh x
or shjc
cosh x
or chjc
tanh x
or thx
cothjc
or cth x
sech x
cosech x
or csch x
Pronounce
sinsh
cosh
tansh
cotansh
secansh
cosecansh
Read
"Hyperbolic sine of x"
"Hyperbolic cosine of x"
"Hyperbolic tangent of x"
"Hyperbolic cotangent of x
"Hyperbolic secant of x"
"Hyperbolic cosecant of x"

Section 14.0 Introduction
535
y
P(cos0, sin0)
A{6) = (1/2) 6
e = 2A
P(cosh t, sinh t)
A(t) = (1/2)*
t = 2A
x
x2+y
2 = 1
Xl-yl = 1
2_
The appellation hyperbolic functions is due to a comparison
between the Pythagorean identities sin2 6 + cos2 0=1 for
trigonometric, or circular, functions and cosh21 - sinh21 = 1
for hyperbolic functions:
o While cos 0 = x and sin 0 = y are the parametric equations
of the circle x2 + y2 = 1, cosh * = x and sinh * = y are the
parametric equations of one branch of the hyperbola x2 -y2 = 1.
o In the trigonometric case, 9 is a radian measure of Z POQ,
which represents twice the area of the sector POQ. In the
hyperbolic case, t is not a measure of an angle; yet, with
the help of integral calculus, t can be shown to represent
twice the area of the hyperbolic sector POQ.
Hyperbolic functions have applications in science and
engineering, where they may express gradual absorption or decay,
e.g., of light, sound, electricity, or radioactivity. They are of
importance for finding integrals.

536 Chapter 14 HYPERBOLIC FUNCTIONS
14.1 Fundamental Hyperbolic Functions
Hyperbolic functions, or fundamental hyperbolic functions,
are generated by the function e*. Their names and notations
are explained and justified by analogies between hyperbolic
functions and trigonometric functions.
The hyperbolic sine and cosine are defined as
e* - e~x _ e* + e~x
sinh x = ; cosh x = ,
where e is the base of natural logarithms and x represents a
real number. Analogously, we have
sinhx ex - e~x e2x-l
tanhx =
cothx =
sech x =
cosech x
cosh x ex + e~x e2x + 1 '
cosh x ex + e~x e2x + 1
ex
e-*
qX + e-x
ex
ex
ex
+
—
2
+
e-^
e-*
e-^ '
2
sinh x ex - e~x e2 x - 1 '
1
cosh x
sinh x e* - e~x
Hyperbolic vs. Trigonometric Functions
The connections between hyperbolic and trigonometric
functions become evident if we compare
pi X __ ci—1 X "^
sin x =
2i
pi-
sinh i x =
e1 x — e-1 x
which gives
sinh i x = i sin x ; sinh x = - i sin i x
and, by analogy,
cosh i x = cos x ; cosh x = cos i x
tanh i x = i tan x ; tanh x = - i tan i x
coth i x = - i cot x ; coth x = i cot i x .
p. 507 Euler*s formula, elx = cosx + isinx, has its hyperbolic
analogue,
ex = cosh x + sinh x .

Section 14.1 Fundamental Hyperbolic Functions
537
Graphs
Graphs of sinh x and coshx may be constructed by combining
the graphs of j- e* and j* e~x'> seen* as l/coshx; cosechjc as
1/sinh x; tanh x as sinh jc/cosh x; and coth x as cosh jc/sinh x.
sinh x
x
x
-3 -2\-l
-1
-2
-3
sinh x
cosech x
x
coihx
r
3
#

Chapter 14 HYPERBOLIC FUNCTIONS
The Catenary
Latin, catena, "chain"
HUYGENS Christiaan
(1629 -1695)
BERNOULLI Jakob
(1654-1705)
A uniform, non-elastic, and infinitely flexible cable or wire
suspended from two fixed points under the influence of gravity
forms a catenary.
The catenary bears great similarity to the parabola and, in
fact, Galileo asserted that what later became known as the
catenary was indeed a parabola. By applying laws of statics
and conceiving the catenary as a succession of equal masses
connected by weightless cords of equal length, Huygens could,
in 1646, refute Galileo's assertion, and thus demonstrate that
the catenary is not a parabola. In the 1690s Jakob Bernoulli
determined the equation of the catenoid.
The catenary may be expressed in the form of a hyperbolic
cosine function,
X ( e*
fix) = a cosh— = a
a
+ e~
where a is a positive constant equal to the y-intercept:
10
Oh (N
a y = f(x) = a cosh ix/a)
*~ X
Catenoid
Minimal Surface
Directrix: p. 406
A catenoid is the surface generated when a catenary is rotated
about its directrix. The catenoid is a minimal surface,
meaning that it takes the shape of least area when bounded by a
given closed space. Soap film between two empty circular
rings takes the shape of a catenoid.
Catenoid

539
14.2 Inverse Hyperbolic Functions
Area Function
There are six inverse hyperbolic functions, or area functions,
one corresponding to each of the six fundamental hyperbolic
functions.
arsinh x (or arsh x)
arcosh x (or arch x)
artanh x (or arth x)
arcoth x (or arcth x)
arsech x
arcsch x (or arcosech x)
Read
Inverse hyperbolic sine of x.
Inverse hyperbolic cosine of x.
Inverse hyperbolic tangent of x.
Inverse hyperbolic cotangent of x.
Inverse hyperbolic secant of x.
Inverse hyperbolic cosecant of x.
Inverse hyperbolic functions are sometimes denoted sinhr1 x,
cosh-1 x, etc.; this form of notation should be discouraged
because of possible confusion with the expression 1/sinh x, etc.
Another form of notation, arcsinh x, arccosh x,etc, is a
practice to be condemned as these functions have nothing
whatever to do with arc, but with area, as is demonstrated by
their full Latin names,
arsinsh
arcosh
area sinus hyperbolicus
area cosinus hyperbolicus, etc.
p. 537
Graphs of hyperbolic functions show that sinh, tanh, coth, and
cosech are odd functions which are one-to-one; cosh and sech
are even functions and are thus not one-to-one. To compute
inverse functions of cosh and sech, their domains must be
restricted, generally by allowing only positive arguments,
x > 0.
With this restriction, we obtain the following graphs of inverse
hyperbolic functions.
arcsch x
arsinh x
i—+~X

540
Chapter 14 HYPERBOLIC FUNCTIONS
arsech x
arcosh x
arcoth x
x
12 3
artanh x

541
14.3 Identities
As with trigonometric identities, hyperbolic identities are
powerful media in calculus.
Fundamental Hyperbolic Functions
Fundamental hyperbolic identities may be transcribed from
trigonometric identities, sinh and cosh replacing sin and cos,
and plus or minus sign being changed before any product of
two sines. The rule can be extended to the remaining funda-
sinh x cosh x
mental hyperbolic functions, tanh x = r—, coth x = . , —,
J cosh x sinh x
1., 1
p. 507
sech x =
cosh x
, and cosech:* =
sinh x
If desired, the reader may check the following identities by
transcribing them from the corresponding trigonometric
identities, or just consult them when needed.
Negative Arguments
sinh (-x)
tanh (-x)
sech (-x)
-sinh x ;
-tanh x ;
sech x ;
cosh (-x) = cosh x
coth (-x) = -coth x
cosech (-x) = -cosech x
Pythagorean Identities
cosh2 x - sinh2 x =
ex + e~*\2 (%x — e~*x2
e2x + 2 + e~2x e2x-2 + e~2x
from which
and, similarly,
cosh2 x - sinh2 x = 1,
seen2* + tanh2:* = 1,
coth2 x - cosech2 x = 1.
Sums and Differences
sinh x ± sinh y =
cosh x + cosh y =
cosh x - cosh y =
tanhx ± tanhy =
coth:* ± cothy =
x±y x + y
2 • sinh —-— • cosh —-—
x + y
2 • cosh —-— • cosh
2 • sinh —-— • sinh
sinh (x ± y)
cosh x • cosh y
sinh (x ±y)
sinh x • sinh y
2
x-y

542 Chapter 14 HYPERBOLIC FUNCTIONS
Products
sinh x • sinhy = -r [cosh (x + y) - cosh (x - y)]
coshx • coshy = — [cosh (x + y) + cosh (x - y)]
sinhx • coshy = r- [sinh (x + y) + sinh (x - y)]
tanh x • tanh y =
2
tanh x + tanh y
coth x + coth y
Double Arguments
2 tanh x
sinh 2 x = 2 cosh x • sinh x ; tanh 2 x =
cosh 2 x = cosh2 x + sinh2 x ; coth 2 x
1 + tanh2 x
coth2x + 1
2 coth x
Half Arguments
• , * if*>0 ^ /cosh x - 1
sinh- = -y g =
sinh x
, x ^ /cosh x + 1 if v >,
cosh- = ^ 2 =
a/2 (coshx + 1)
>0 sinh x
V2 (coshx- 1)
x sinh x ifx*0 cosh x -
2 "" cosh x + 1 = si
shx - 1 ifjc>o A /cosh x - 1
inh x = \ cosh x + 1
,, £ ifjc^O sinhx jfjc^o coshx + 1 jfjoQ /cosh x + 1
2 = cosh x - 1 = sinh x = V cosh x - 1
Multiple Arguments
sinh 3 x = sinh x (4 cosh2 x - 1)
sinh 4 x = sinh x • cosh x (8 cosh2 x - 4)
sinh 5 x = sinh x (1 - 12 cosh2 x + 16 cosh4 x)
cosh 3 x = cosh x (4 cosh2 x - 3)
cosh 4 x = 1 - 8 • cosh2 x + 8 cosh4 x
cosh 5 x = cosh x (5 - 20 cosh2 x + 16 cosh4 x)
sinh n x = ( -j ) • cosh" ~ 1 x • sinh x + ( o ) • cosh" ~ 3 x • sinh3 x
+ ( k ) • cosh" ~ 5 x • sinh5 x + ...
cosh n x = cosh" x + ( 2 ) • cosh" ~ 2 x • sinh2 x
+ (4)- cosh" " 4 x • sinh4 x + ...

Section 14.3 Identities
543
Compound Arguments
sinh (x ± y) = sinh x • cosh y ± cosh x • sinh y
cosh (x ± y) = cosh x • cosh y ± sinh x • sinh y
tanh x ± tanh y
tanh(%±y) =
coth (x ± y) =
1 ± tanh % • tanh y
1 ± coth x • coth y
coth x ± coth y
Powers of Arguments
sinh2 x = o" (cosh 2 % - 1)
cosh2* = o" (cosh 2 % + 1)
sinh3 x = j (- 3 sinh x + sinh 3 *)
cosh3 x = j (3 cosh % + cosh 3 x)
sinh4 * = jr (3 - 4 cosh 2 x + cosh 4 *)
cosh4 x = g (3 + 4 cosh 2 % + cosh 4 *)
sinh5 x = tt (10 sinh % - 5 • sinh 3 x + sinh 5 x)
coSh5, = ^(10coSh, + 5.coSh3x + cosh5x)
sinh6:r = oq" (- 10 + 15 • cosh 2 % - 6 cosh 4 % + cosh 6 *)
cosh6* = 09" (10 + 15 • cosh 2x + 6 cosh 4% + cosh6x)
Complex Arguments
sinh (x ± i y) = sinh % • cos y ± i • cosh x • sin y
cosh (x ± i y) = cosh % • cos y ± i • sinh x • sin y
sinh 2 % ± i • sin 2 y
tanh (x ± i y) =
coth Or ± i y)
cosh 2 % + cos 2 y
sinh 2 * + i- sin 2 y
cosh 2 % - cos 2 y
sin (% ± i y) = sin x • cosh y ± i • cos x • sinh y
cos (x ± i y) = cos % • cosh y + i • sin x • sinh y
. . . sin 2 % ± i • sinh 2 y sin 2 % ± i • sinh 2 y
tan (% ± 1 y) = - r-r = 0 -
cos 2 % + cosh 2 y 2 (cos2* + sinh2 y)
sin 2 x + i • sinh 2 y sin 2 * + i • sinh 2 y
cos 2 * - cosh 2 y "2 (sin2* + sinh2 y)
sinh (* + i • 2 n n) = sinh *
cosh (x + i • 2 n n) - cosh *
tanjh (x + i-nn) = tanh *
coth (* + i • n n) = coth *
cot (* ± i y) = -

544
Chapter 14 HYPERBOLIC FUNCTIONS
Conversion Table
QinVi v —
Olllll wv —
cosh* =
tanhx =
prwrti v —
1/Uld.l A> —
sinh
—
Vl + sinh2 x
sinh x
Vl + sinh2 x
Vl + sinh2 x
sinh x
cosh
+ Vcosh2 x -1
Vcosh2 x - 1
cosh #
cosh #
Vcosh2 #-1
tanh
tanh x
Vl - tanh2 x
1
Vl - tanh2 x
1
tanh #
coth
1
H
V coth2 x - 1
| coth # |
V coth2 x - 1
1
coth #
Inverse Hyperbolic Functions
sinh (arsinh x) = arsinh (sinh x) = x
cosh (arcosh x) = x; arcosh (cosh x) = x, if* > 0
Sums and Differences
arsinh x ± arsinh y = arsinh \x • V 1 + y2 ± y • V 1 + x2)
arcosh x + arcosh y = arcosh \x y + V(*2 - 1) (y2 - 1))
arcosh x - arcosh y = arcosh \x y - V (x2 - 1) (y2 - 1))
• sgn (x -y),
where sgn (x -y) refers to the sign of the difference
(that is, 1 or - 1)
x ±y
artanh x ± artanh y = artanh
arcoth x ± arcoth y = arcoth
1 ±xy
l±xy
x ±y

Section 14.3 Identities
545
Conversion Table
arsinh
arcosh (x > 0)
artanh
arcoth
arsinh x =
arcosh x =
artanh x =
arcoth x
arsinh V*2 - 1
arsinh
x
arsinh ±
l
V^T
arcosh Vx2 + 1
arcosh
arcosh
Vi-^2
X
artanh
artanh
x
Vx2 +1
X
artanh —
x
arcoth
arcoth
Vx2 +1
x
x
Vx2-!
arcoth —
x
Inverse Hyperbolic Functions vs. Natural Logarithms
Since hyperbolic functions are combinations of exponential
functions ex, we may expect simple relations to exist also
between inverse hyperbolic functions and natural logarithms.
If
y = arsinh x,
where
we have
x = sinhy = — (e^ - e~^) ,
e^ -e~y = 2x
(eyf-2x-ey -1 = 0
ey = x (+) Vx2 + 1,
where the minus sign is discarded, since e^ > 0. The natural
logarithm of both members of the square gives
[y =] arsinh x = In \x + Vx2 + l) .
Similarly, we find:
arcosh x = In \x + Vx2 - 1 ) ;
artanh x = — • In ;
2 1 - x
1 _ x + 1
arcoth x = — In
arsech x = In
2 x- 1 '
i + Vi-*2
x
arcsch x =
In
In
1 + Vl +x2
X
1 + Vl +x2
X
X>1
\x\ <1
\x\ >1
l>x>0
x > 0
sgn x ; x ?t 0,
where sgn x refers to the sign of x (that is, 1 or - 1)

547
Chapter
15
ANALYTIC GEOMETRY
Page
15.0 Scope and History 548
15.1 Rectilinear Figures 549
15.2 Conic Sections 559
15.3 Shifting Orthogonal Coordinates 572
15.4 Polar Coordinate Systems 576
15.5 Parametric Equations 586

548
Chapter 15 ANALYTIC GEOMETRY
15.0 Scope and History
Coordinate Geometry
APOLLONIOS
(255 -170 B.C.)
ARCHIMEDES
(287-212 B.C.)
DESCARTES Rene
CARTESIUS
(1596-1650)
p. 367
de FERMAT Pierre
(1601-1665)
NEWTON Isaac
(1643 - 1727)
von LEIBNIZ Gettfried Wilhelm
(1646 - 1716)
In analytic, or coordinate, geometry, geometric problems are
made accessible to algebraic reasoning by introducing the
connection between points and numbers - the fundamental
elements of geometry and algebra, respectively. Once a
coordinate system is defined, every point of a plane curve is
uniquely represented by an ordered pair of real numbers,
every point of a three-dimensional surface by an ordered
triplet of real numbers.
By the 3rd century B.C., Greek mathematicians Apollonius of
Perga and Archimedes had already used longitude, latitude,
and altitude to define the position of a point, but it was not
until the 17th century that the French mathematicians Rene
Descartes and Pierre de Fermat directed this idea into
systematic use.
Descartes - Latinized Cartesius - presented the fundamental
concepts of coordinate geometry in La geometrie (1637), which,
with Newton's Principia, is one of the most influential
scientific texts of the 17th century. By representing a point by a
pair of real numbers, and straight lines and curves by
equations, Descartes provided a link between geometry and
algebra. Yet, rather than devising a system for algebrizing
geometry, Descartes's purpose was to advance the methods of
geometric construction.
Fermat had outlined the principles of analytic geometry even
before the publication of La geometrie, but his text was
circulated for many years among mathematicians in
manuscript form until it was finally printed, in 1679, in Varia opera
mathematica as "Ad locos pianos et solidos isagoge"
("Introduction to Plane and Solid Loci"). In addition to being one of
the originators of analytic geometry, Fermat - a lawyer by
profession - was the founder of modern number theory and,
with Blaise Pascal, of probability theory; Fermat, "the king of
amateurs", also devised and applied the principal idea of
differential calculus.
Descartes and Fermat did not consider negative values for
distances and, consequently, their works contain no
orthogonal Cartesian coordinate systems in the modern sense. The
concept of negative distances is due to Newton and Leibniz.
Newton is also considered the originator of polar coordinates,
where a point is defined by a pair of real numbers, one
representing its distance from a fixed point, the origin, the other an
angle between a fixed ray and a radius from the origin to the
given point.
The development of analytic geometry and calculus in the
17th century signaled the beginning of modern mathematics.

549
15.1 Rectilinear Figures
y
A
Second Quadrant 4-- First Quadrant
3
2
Q(-4,l) • i-
H 1 1 h
-4 -3 -2 -1
Third Quadrant
-1--
-2--
-3--
-4--
0
P(2, 3)
1 1 1 !-► X
12 3 4
Fourth Quadrant
jc-axis
y-axis
Origin
Abscissa
Ordinate
A point in the plane can be located in an orthogonal
(rectangular, or Cartesian) coordinate system formed by two
perpendicular real number lines, the x-axis and the
y-axis. The point of intersection of the axes is called the
origin, denoted O. Each point in the plane is determined by an
ordered pair of real numbers (x, y). The axes divide the plane
into four quadrants.
The ^-coordinate, or abscissa, of a point is its horizontal
distance from the y-axis measured parallel to the x-axis, and is
positive if the point lies to the right of the y-axis and negative if
it lies to the left of the y-axis. The abscissa of P is 2, of Q is - 4.
The y-coordinate, or ordinate, of a point is similarly defined to
be its vertical distance from the x-axis measured parallel to
they-axis. The ordinate of P is 3, of Q is 1.
A point in space can be located by determining its distance to
three perpendicular planes, intersecting at the origin. Any
such planes intersect in a line and, consequently, the
coordinate system in space has a total of three coordinate axes,
usually denoted x, y, and z:
z
/
V,xtytz)

550
Chapter 15 ANALYTIC GEOMETRY
Distance Formulas
In the Plane
In a two-dimensional orthogonal coordinate system, the
distance between point (x\, y\) and point (#2, y?) is
d = V (x2 - *i)2 + (y2 - Ji)2
Why?
y
A
-^1
^2-^1
-½
(«i»yi)
(*2, y2)
A-1
Xf
#
2 1
By the Pythagorean theorem, we find
d2 = (x2-xi)2 + (y2-yi)2
d = V (x2 - *i)2 + CV2 - y i)2 •
In Space
In a three-dimensional orthogonal coordinate system, the
distance between point (xi,yi,zi) and point 0¾ ^2^2) is
d = V (*2 - *l)2 + Cv2 - y l)2 + (22 ~ 2l)2 •
Why?
z
A
v^l > ^1 > ^1
v^2> ^2» Z2'
X
These triangles have one side in common, the length of which,
with Pythagoras, is
Va2 + b2 .

Section 15.1 Rectilinear Figures
551
The sought distance d is the hypotenuse of a right triangle
whose other sides are ^a2 + b2 and c; using the Pythagorean
theorem, we obtain
d2 = (Va2 + 62)2 + c2
d = V a2 + b2 + c2 .
Since a = %2-x\\ b=y2~yi', c = z^-zi, we obtain
d = V (x2 -*i)2 + (^2- yi)2 + («2 -^l)2 .
Midpoint of a Line Segment
*p
VC
Whether in the plane or in space, each coordinate of the
midpoint of a line segment is the average of the corresponding
coordinates of the endpoints of the segment; thus,
xM
x<l + X\
yu =
y2 + yi
*M
Z2 + Z1

552
Chapter 15 ANALYTIC GEOMETRY
Area of a Triangle
The formula for the area (A) of a triangle with vertices at
points (xi>yi), (x2,y2)> (^3^3) is
A =
2 ki 0^2 - ys) - *2 (y 1 - ^3) + *3 (y 1 - ^2)]
Why? Atnangle = 2 (base x height)
^trapezoid = 9 (sum of parallel sides x height)
Let the vertices of a triangle be P\(x\, yi)> ^2(^2» J2\ ^3(*3> ^3)»
and assume P^ to be the lowermost vertex.
Let Qi and (¾ be projections on a line parallel to the x-axis and
passing through P2:
pibi,yi)
Ql (^1,^2)
P» (^3^3)
x
P2 (*2> ^2)
^trianglep1p2P3 ~
2{XS
The area of triangle P1P2P3 equals the area of trapezoid
P1Q1Q2P3 less triangles PiQiP2 and P2Q2P3:
*i) [(yi -^2) + (y3 -^2)] - 2 (*2-^1)(^1-^2) - 2 ^3-^2)(^3-^2)
= 2 ki (^2-^3) + *2 (^3-yi) + *3 (y 1-^2)] •
Instead of being in a point of segment Q1Q2, vertex P2 may be to
the left of Qi or to the right of Q2:
x
x
_**Hi
In the left-hand diagram, the area of triangle P1P2P3 equals
the area of trapezoid P1Q1Q2P3 plus the area of triangle P1Q1P2
less the area of triangle P2Q2p3-

Section 15.1 Rectilinear Figures
553
In the right-hand graph, triangle P1P2P3 equals the area of
trapezoid ^iQiQ2^3 plus the area of triangle P3Q2P2 less tne
area of triangle P\Q\P2-
These two cases also lead to
^trianglep1p2P3 = 2 ^1 ^2 ~J^ + *2 ^3 ~Jl) + *3 ^1 ~ ^ '
If we use the absolute value of the result,
A =
2 [*i (y2 - ys) - *2 Cvi - ^3) + *3 Cvi - ^2)]
the vertices of the triangle may be read in any order.
The formula expressing the area of a triangle in terms of the
coordinates of its vertices may be applied to find the area of
any polygon.
Equations and Graphs of Straight Lines
y
A
y = b
(a,b)
•x
X
Horizontal and vertical lines may be identified by their
constant y-coordinates and x-coordinates, respectively.
A straight vertical line through point (a, b) is identified by the
equality
x = a;
a straight horizontal line through (a, b) by the equality
y = b .
Slope, Inclination
"Rise over Run"
Point-Slope and Two-Point Equations
The slope, or inclination, of a line is defined as the ratio of the
vertical distance to the horizontal distance from any one point
to any other point on the line. This ratio is sometimes
conveniently referred to as "rise over run".
A straight line that has the slope m and contains the point
(xi, y\) has the point-slope equation
y -yi = m (x-x\) .

554
Chapter 15 ANALYTIC GEOMETRY
Consider two points (xi, y\) and 0c2, y2) on a line. The slope m
is calculated as the ratio of Ay = y2 -y\ units in the vertical
direction and Ax = x2 - *l units in the horizontal direction,
where A denotes an increment:
y
A
y<i
y\
(*2> y2)
^^T)
^*$/\a A\ riSe
| x2 - xx i
' run i
i i
X1 #2
X
m = tan a =
rise
run
Ay
Ax
y2-yi
x2 -*i
; *i * *2
If we know two points (*i, yi) and (x2, y2) of a straight line, the
line has the two-point equation
y -yi = vac -x\).
Slope-Intercept Equation
If a straight line is not parallel to the y-axis, it must intersect
they-axis at some point (0, 6), called the ^-intercept of the line:
**- x
As point (0, b) is on they-axis, the point-slope equation,
y -yi = mix - x\), may be simplified to the slope-intercept
equation
(y - b) = ra (x - 0),
or u
y = rax + b .

Section 15.1 Rectilinear Figures
555
The equation of a line with the slope m = - ■=■ and a point (- 6, 9)
is
y-9 = -- [x-{-6)]
_ 1 39
y ~~5X+5 '
General Form of the Equation for a Line
If the slope-intercept form of a line
y = mx + b
is rearranged in a form where
A C
m is-— and b is - — ,
we obtain the general form, or standard form, of the equation
for a line,
Ax + By + C= 0.
Find the slope and the y- and x-intercepts of a line
4x+2y-8 = 0.
To find the slope m and the y-intercept 6, we have
4 -8
m = --= -2; 6 = -—= 4.
The x-intercept is determined by setting y = 0,
4x = 8; x = 2 .
Parallel or Perpendicular Lines
Parallel lines have the same slope.
Lines
L\\ 3y = 5x + 4 and L2 : 3 y = 5 x + 9 are parallel
but
Li: 2y = 3x + 4 and L2: y = 3 x + 4 are not.
Perpendicular lines have slopes that are negative reciprocals
of one another.
Lines
L\\ 3y = 5x + 4 and L3: 5 y = -3 x + 9 are perpendicular
but
L\\ 3y = 5x + 4 and L2'. 3y = 5x + 9 are not.

556
Chapter 15 ANALYTIC GEOMETRY
Intersecting Lines and Their Angles
The angle between two intersecting straight lines is defined as
the smallest positive angle through which the first line must
rotate to coincide with the second line.
The angle between two lines may be determined from the
values of the slopes of the lines.
Two types exist. In the figures below, we have
to the left: <t>L\< $L2
to the right: 0£2 < 0Li
p. 504
x
Using the notations of the figures, the angle B, between two
lines, whose slopes are m\ and ra2, is
and
4 = arctan "2-mi
1 + ra2 mi
| = 180° + arctan f2""11
1 + ra2 mi
if ™*-mi >o
1 + ra2 mi
if r2-mi <o.
1 + rrt2 mi
Why? For ¢^ < <fc2, we have
for <j>L2 < fal9
<S>L2 = <l>Li + £>
5 = <t>L2 ~ <fel ;
<t>Li = 0L2 + (180° -#
£ = 180° + (<fc2 - 0Ll).
Since the tangent function has period 180° (n rad), we have in
either case that
„ , x tan <j>L9 - tan <fci
tan£ = tan(<fc9-0Ll) = -—-2^ ^1
b vvz,2 y^f l + tan <fc2 tan 0Ll '
provided that neither 0Ll nor <fe,2 is 90° (tan 90° is not defined).

Section 15.1 Rectilinear Figures
557
Let the slopes of L\ and L2 be mi and m2, respectively.
As
m\ = tan fai and m2 = tan 0£2,
we have
tan£ =
m2 ~ mi
p. 476
1 + m2 mi
If the quantity on the right is positive, £ is simply its arc
tangent function. If the quantity on the right is negative, then
its arc tangent function is negative, and the positive angle £ is
obtained by adding 180°.
Determine the angles between the intersecting lines
a: L\ : y = 2x + 3 and L2: 2y = -Sx + 8
b: Li: y = 2 and L2: 3y = -x^l~3
a:
Let the slope of lines y = 2 x + 3 and 2 y = ■
rri2, respectively; then,
3
mi = 2 and m2 = -~ .
3 x + 8 be mi and
£ = arctan
-!-*
X
1 +
(412
arctan - & 60.26°
4
b:
mi = 0; rri2 = -
V3
V3
- 0
^ = 180° + arctan
r
= 180° +arctan
1 +
r
\ 3
0
£3^
V 3
= 150c
H 1 1 1 1 1 1 1 1 h
-9
-6
-3
^—i—\-
x

558
Chapter 15 ANALYTIC GEOMETRY
If one of Li, L<i is vertical, the formula does not apply, but the
situation is easy to handle.
Determine the angle between the intersecting lines
L\ : y = x\3 and L2 : x = 2
Z/2 is parallel to the y-axis; the sought angle is equal to the
angle between L\ and the y-axis.
Let the slope of line y = x be mi,
run ^3
rise 1
and let the sought angle be 0,
1
V3
<S> = 30° .
tan0 =
*->> x

559
15.2 Conic Sections
The Circle
Latin, circulus, diminutive
form of circus, "ring", akin to
Greek kirkos, "ring"
General Equation
Central Equation
A circle is the locus (set) of points (x, y), in the plane, that are at
a given distance, called the radius (r), from a fixed point, the
center (u, v):
y
x
Equations of the Circle
The equation of a circle centered at any point in the coordinate
system - the general equation - is obtained by expressing the
radius r in terms of the distance formula; thus,
leading to
- = y (x - u)2 + (y - v)2 ,
(x - u)2 + (y - v)2 = r2 .
The equation of a circle centered at the origin of the coordinate
system - the central equation - is
x2+y2 = r2
A circle centered at (3, - 5) and with the radius 3 units of length
has the equation
(x- 3)2 + {y + 5)2 = 9.
The Ellipse
Greek, ellipsis, "a defect",
indicating that the angle
between the ellipse and the base
of the cone is less than the angle
between the parabola and the
base of the cone
An ellipse is the locus of points in the plane for which the sum
of their distances from two fixed points (foci) is constant.
In the graph below Fi and F2 denote the foci of the ellipse;
P represents any point on the ellipse.
The sum of the distances | Fi P \ and | F2 P \ is constant; this
constant value equals the length of the major axis, 2 a.
P to, y)

560
Chapter 15 ANALYTIC GEOMETRY
Equations of the Ellipse
Central Equations
Central equations concern an ellipse whose axes meet at right
angles at the origin of the orthogonal coordinate system.
Two cases occur: the one for the ellipse whose major axis
coincides with the %-axis, the other for the ellipse whose major axis
coincides with the y-axis:
- + ^-= 1
2 b2
a
xi
b2
= 1
ad
center at the origin, major axis
coinciding with the %-axis;
center at the origin, major axis
coinciding with the y-axis,
General Equations
where 2a and 26 are the axes of the ellipse.
If, instead, the center of the ellipse is at a point (u,v), we have
the general equations:
(x - u)2 (y - v)2 major axis parallel to or
a4
b2
coinciding with the %-axis;
(x - u)2_ (y - v)2 major axis parallel to or
b2
a'
coinciding with the y-axis.
To deduce the central equations, let P(x, y) represent any point
on the periphery of an ellipse centered at the origin and major
axis 2 a coinciding with the %-axis of an orthogonal coordinate
system; let the minor axis of the ellipse equal 26, and the
distance between the center and a focus (F\ or F2) equal c.
y
t
P (x, y)
( J 0
W<-*.o>
- c ~
a
~""-,,-^v^ t
^^^-a f ^
^i (c, oy
c
< a
i
1
k
6
t
6
'
The distance formula gives
|FlP| = V (x - c)2 + (y - 0)2
and
x
\F2P\ = V [* _ (-c)] 2 + (y-0)2 .

Section 15.2 Conic Sections
561
By definition of the ellipse, the sum of the distances | F\ P | and
i<2 P | is equal to the length 2a of the major axis; thus,
c)]2 + (y - 0)2 = 2 a,
which may be rewritten
V (x - c)2 + y2 = 2a - V (* + c)2 + y2 .
Square both sides,
x2 - 2 c x + c2 + y2 = 4 a2 - 4 a \ Gc + c)2 + y2 + x2 + 2 c x + c2 + y2,
and simplify,
a \ Oc + c)2 + y2 = a2 + ex.
Again, square both sides and simplify,
a2 x2 + 2 a2 c x + a2 c2 + a2y2 = a4 + 2 a2 c x + c2 x2;
(1) (a2-c2)jc2+ a2y2 = a2(a2-c2).
Since the sum of | Fi P \ and | F2 P \ equals the length of the
major axis, 2a, the distance between either focus and an end
point of the minor axis is a.
y
k
P(x,y)
x
By the Pythagorean theorem:
62 + c2 = a2
a2-c2 = b2
Substituting b2 for a2 - c2 in (1) gives
b2x2 + a2y2 = a2 62
and, after dividing both sides by a2b2, we find the central
equation

562
Chapter 15 ANALYTIC GEOMETRY
For the general equation of an ellipse whose major axis
coincides with or is parallel to the %-axis of the coordinate
system, consider the graph of the ellipse below, whose center is
at a point {u, v):
y
A
^ a
x
By definition of the ellipse, we have
^1 [x-(u-c)]2 + (y-v)2 + ^1 [x-(u + c)]2 + (y-v)2 = 2a
which, analogously to the central equation, may be simplified
and rewritten
(x - u)2 (y - v)2
a*
b2
= 1.
To derive the central and general equations of an ellipse
whose major axis coincides with or is parallel to the y-axis,
one follows a similar pattern.
The Parabola
A parabola is the locus of points (x, y), in the plane, whose
distances to a given straight line (the directrix) and from a
given point (focus) outside the line are equal, as illustrated by
the distances d\ and (¾ below:
x
The parabola is symmetrical with respect to an axis passing
through the focus and perpendicular to the directrix. The
vertex of the parabola is defined to be located at the midpoint of the
segment of the axis extending from the focus to the directrix.

Section 15.2 Conic Sections
563
Central Equations
General Equations
Equations of the Parabola
Central equations concern a parabola whose axis coincides
with one of the axes of the coordinate system and whose vertex
is at the origin. Two cases occur: the one for the parabola
whose axis coincides with the y-axis, the other for the parabola
whose axis coincides with the x-axis:
axis coinciding with the y-axis;
y* = 4px axis coinciding with the x-axis,
where p is the distance between the vertex and the focus.
x2 = Apy
'2 - 4px
If, instead, the vertex of the parabola is at a point (u, v), we have
the general equations:
(x-u)2 = Ap(y -v) vertex at (u, v), axis parallel to or
coinciding with the y-axis;
(y - v)2 = 4p (x-u) vertex at (u, v), axis parallel to or
coinciding with the x-axis.
To deduce the central equations, let P(x, y) be a point on the
parabola with its vertex at the origin, axis coinciding with the
y-axis, and focus at a distance p from the vertex (p > 0):
J
I
>v (0,p\
directrix
(b, -p)
/
[
^-
7
. n
*
(x,
y
\
-P)
X
The directrix isy = -p.
di = V(*-0)2 + (y-p)2 ; d2 = y - ( -p)
By the definition of the parabola, d\ = d<i\ thus,
(1) V(*-0)2 + (y-p)2 = y -( -p)
•2 - ^py .
X'
For the general equation of a parabola whose axis is parallel to
or coincides with the y-axis, consider the graph of the parabola
below, whose vertex is at {u, v):
x

564
Chapter 15 ANALYTIC GEOMETRY
By definition of the parabola, we have
V (x - u)2 + \y - {v + p)]2 = y - (v -p)
(x - u)2 = Apiy -v) .
For the central and general equations of a parabola whose axis
coincides with or is parallel to the x-axis, one follows a
similar pattern.
The Hyperbola
A hyperbola is the locus of points, in the plane, for which the
difference of their distances from two fixed points (foci) is
constant.
y
i
P(x,y)
x
In the above graph, where F1 and F2 denote the foci of the
hyperbola, the difference of the distances d± and d2 from each
focus to any point on the curves of the hyperbola is constant.
transverse
axis
Referring to the above graph of the hyperbola, we give the
following definitions:

Section 15.2 Conic Sections
565
Vertices
Transverse Axis
Major Axis
Center
Conjugate Axis
Asymptote
Greek, asymptotos,
"not falling together"
Minor Axis
Principal Axis
o The line of indefinite length through the two foci and
intersecting the vertices (Vi, V2) is the transverse axis.
0 The segment of the transverse axis between the vertices is
the major axis.
0 The midpoint of the major axis is the center (C).
0 The line of indefinite length through the center and
perpendicular to the transverse axis is the conjugate axis.
0 An asymptote is a straight line which, indefinitely
produced, does not meet the hyperbola but comes arbitrarily
closer to it. The asymptotes of the hyperbola intersect at its
center.
0 The segment of the conjugate axis that has the same length
as a line parallel to the conjugate axis, connecting the
asymptotes and one vertex, is the minor axis.
The terminology may vary from text to text. Thus, another
term for transverse axis is principal axis. The terms
transverse axis and conjugate axis are in some texts used for
the line segments that in the above are referred to as major
axis and minor axis, respectively.
Equations of the Hyperbola
Central Equations
General Equations
Central equations concern a hyperbola whose axes intersect at
the origin of the coordinate system. Two cases occur: the one
with the major axis coinciding with the x-axis, the other with
the major axis coinciding with the y-axis.
major axis = 2a,
minor axis = 26,
b = ycP^a2, where c is the distance between the center
and a focus of the hyperbola,
then the central equations of the hyperbola are
X'
a-
y
a'
b2
b2
= 1
= 1
center at the origin,
foci on the x-axis;
center at the origin,
foci on the y-axis.
If, instead, the center of the hyperbola is at a point (u, v), we
have the general equations:
(x - u)2 (y - v)2
a'
b2
(y - u)2 (x - v)2
a4
b2
= 1
= 1
foci on a line parallel to or
coinciding with the x-axis;
foci on a line parallel to or
coinciding with the y-axis.

Chapter 15 ANALYTIC GEOMETRY
To deduce the central equations, let P(x, y) represent any point
on a hyperbola centered at the origin and with foci on the
jc-axis.
Using the distance formula and letting the distance between
center and focus {F\ or F%) be c, then:
x
d1 = V (x - c)2 + (y - 0)2 = V(*-c)2+y2
d2 = V [x - (-c)] 2 + (y-0)2 = V (x + c)2+y2
From the definition of the hyperbola, we have that the
difference between P{x, y) and the foci equals the length of the
major axis, 2a; thus,d^-d\ = 2a, or
V (* + c) 2 + y2 - V (x - c)2 + y2 = 2a ,
rewritten
\ (x + c)2 + y2 = 2 a - \ (x - c)2 + y2 .
Square both sides,
x2 + 2 c x + c2 + y2 = 4 a2 - 4 a \ (jc - c)2 + y2 + #2 - 2 c # + c2 + y2,
and simplify,
a\ (*-c)2+y2 = ex-a2.
Again, square both sides and simplify,
a2 x2 - 2 a2 c x + a2 c2 + a2y2 = a4 - 2 a2 c # + c2 x2;
(1) (c2-a2) jc2-a2y2 = a2(c2-a2).
Referring to the figures, we obtain a2 + b2 - c2, and can define
b = Vc2 - a2 ;
after substituting 62 for c2 - a2 in (1), we obtain
b2x2 — a2y2 = a2 b2
and, after dividing both sides by a2 b2,
X'
ai
62
= 1
For the general equation of a hyperbola with foci on a line
coinciding with or parallel to the jc-axis, consider the graph of
the hyperbola below, centered at a point (u,v):
x

Section 15.2 Conic Sections
567
Fi(-3,0)
From the definition of the hyperbola, we have
yj [x - {u - c)] 2 + (y -v)2 + yj [x - {u + c)] 2 + (y - v)2 = 2a ,
which, after some algebra, may be rewritten
(x - u)2 (y - v)2
a-
b2
= 1.
A hyperbola is centered at the origin and has its transverse
axis coinciding with the x-axis of an orthogonal coordinate
system; the distance between the center and a vertex is 2; the
distance between the center and a focus is 3.
Determine the equations of the hyperbola and its asymptotes.
Since the hyperbola is centered at the origin, we use the
equation
x-
a'
b2
= 1,
where
Vi(-2,0\\ //V2(2,0)
F2 (3, 0)
x
major axis = 2 a
minor axis = 2b
b = V c2 -a2
c is the distance between the center and a focus of
the hyperbola.
6 = Vc2-a2 yields b2 = 32-22; 6=^5.
The equation of the given hyperbola is
_
4
r
= l.
x
Written in slope-intercept form, the equations of the
asymptotes of a hyperbola, centered at the origin and with its major
h h
axis coinciding with the x-axis, are y = —x; y = - — x , and
a
a
for the given hyperbola
V5
2
y = —* ; y =-—* •
2

568
Chapter 15 ANALYTIC GEOMETRY
The Vertex Equation
Directrix
Eccentricity
The general definition of a conic section is:
The locus of all points in the plane whose distances to a
given point, the focus, and to a given straight line, the
directrix, in the same plane, are in a specific, constant
ratio, known as the eccentricity.
As might be expected, the equations of the conic sections can all
be brought into a common form.
Kftf''
<g$r hyperbola
parabola
ellipse
x
x

Section 15.2 Conic Sections
569
The central equation is
for the ellipse
X2
a2 "
where
2a
26
2c
a2
d
D1D2 =
FiF2 =
V1V2 =
y2
the major axis
the minor axis
the interfocal
distance
62 + c2
the distance
between the center
and the directrix
the directrices
the foci
the vertices
for the hyperbola
2a
26
2c
a2
d
D1D2
F1F2
VXV2
X2
a2
—
=
—
=
•^■t
=
=
=
y2
62 " 1
the transverse
axis
the conjugate axis
the interfocal
distance
c2 -62
the distance
between the center
and the directrix
the directrices
the foci
the vertices
The general definition applies for all points on the curves;
specifically for the vertices, we have the distances:
for the ellipse for the hyperbola
vertex V\ to focus F\ ,
a - c
vertex V\ to directrix D\,
d - a
vertex V2 to focus Fi,
a + c
vertex V2 to directrix D\,
d + a
We then have the eccentricity,
a -c
a + c
e =
d - a d + a
or
ad + a2 - cd - ac
= ad + cd - a2 - ac
a2 = cd:
and d =
a'
which insert in
a + c a + c
e =
d + a
ad
c (a+ c)
a (a + c)
+ a
c - a
a-d
c +a
a + d
c -a
c + a
e =
a-d d + a
or
ac - a2 + cd - ad
= ac + a2 - cd
2 = cd;
-ad
a
7 a
and d = —
c
e =
c+a c + a c (a+ c)
a+d
a
2 a (a + c)
a+
and
= cla .
and
= cla .

570 Chapter 15 ANALYTIC GEOMETRY
We introduce in the central equations the point P,
*ellipse = v"c > P) > -^hyperbola = ^c > Ph
Latex Rectus, Focal Chord where 2p is the latex rectus (plural: latera recta), or focal
chord,
(-c)2 jo^ c2 p2
a2 b2 ' a2 62
or
b2 (a2 - c2) rt>2>>2 62 (c2 - a2) /fc2A2
P ~ a2 ~{a) ' P ~ a2 ~{a
and
62 62
r a r a
Shift the curve a distance a in the x direction so that one vertex
will be at the origin,
the ellipse toward the the hyperbola toward the
right, left,
(x - a)2 y2 _ (x + a)2 y2 _
a2 + b2 " ; a2 " b2 " ;
x2 2 ax a2 y2 x2^ 2ax a2 y2
a2 a2 a2 b2 ' a2 a2 a2 b2
2 2b2 b2 2 2 2&2 b2 2
a aL a az
y2 = 2p% - (l-e2)*2; y2 = 2px - (l-e2)x2;
that is, the same equation, in vertex form, for the ellipse and
the hyperbola; it also applies for the parabola, with e = 1,
y2 = 2px,
and for the circle, with e = 0 and p = r,
y2 = 2rx - x2;
or x2 - 2rx —y2 = 0
and (x - r)2 + y2 = r2 .
Consequently, the vertex equation
y2 = 2px - (l-e2)x2
describes ^ Greek
word means
for e = 0: a circle kirkos a ring - 1 • %2
0<e < 1 : an ellipse elleipein to deduct - (l-e2)*2
e = 1: a parabola paraballein to equate + 0 • x2
e > 1: a hyperbola hyperballein to exceed + (e2 - 1) %2
Another illustration of the meaning of the names elliptic,
parabolic, and hyperbolic is offered by the Euclidean and non-
Euclidean geometries in the interpretation of the Euclidean

Section 15.2 Conic Sections
571
parallel axiom; through a given point outside a given straight
line a number of other straight lines can be drawn, parallel to
the first line:
- in elliptic geometry less than one, that is, none;
(Riemann)
- in Euclidean geometry just one;
("parabolic")
- in hyperbolic geometry more than one
(Gauss, Bolyai,
Lobachevski)
Beyond the Conies
With n = 2,
X
a
n
+
y
b
is the formula for the ellipse, but with other values of n, we have
The graph of the formula with n = 2.5, once considered an ideal
design for a tabletop, was named "super-ellipse" by Piet Hein,
Danish architect, cartoonist/satirist, and author of Grooks.
With n = 2,
X
a
n
y
b
is the formula for the hyperbola. Examining for other values
of n, we have

Chapter 15 ANALYTIC GEOMETRY
Shifting Orthogonal Coordinates
By imposing a new orthogonal system on the original system,
an equation can sometimes be made more accessible.
We distinguish two forms of transposition:
o Translation: A parallel displacement of the original
system along one or more of its axes.
o Rotation: The origin of the new system coincides with the
origin of the original system.
Translation
If after translation the coordinates x,y, and z of a point P
become x\y\ and z', then the translation equations
x = xx + a y=y' + 6 z = z* +c
imply that a, 6, and c are the coordinates of the origin of the
x\ y', z'-system with reference to the x, y, z-system:
x
x
A circle has the equation (x - 3)2 + (y - 2)2 = 1. After
translation, the origin of the new coordinate system {x\ yl) is
located at (3, 2) with reference to the original (x, y) coordinate
system. Determine the equation of the circle expressed in the
coordinates (x\yl).
The original equation represents a circle of radius 1 and
center at (3, 2) in the original system.
After translation, we have
x = x' + 3;y=y' + 2.
The new equation is
x [te' + 3)-3]2+[(y' + 2)- 2]2 = 1
or
(x'P + iy1)2 = 1.

Section 15.3 Shifting Orthogonal Coordinates
573
Rotation
Expressing New Coordinates in Original Coordinates
With the origin fixed, let the x, y-axes rotate counterclockwise
through an angle 0into a system of x\y'-axes:
A P(x\y')
,fP(x9y)
\
.£&
a
X
y
r cos a
r sin a
x
\
\
\
\
\
\
4t
^y
y
,y
y
(x,y)
/
/
>
„y
y
X
\
\
x' = r cos (a- 6)
y' = r sin (a - 6)
p. 509
Expressed with reference to the original coordinates (x, y), the
new coordinates (x\ yx) are
x' = x cos 6 + y sin 0; y' = - x sin 6 + y cos 6.
x'
Proof: cos(a-0) = —
r
xx = r cos (a- 6).
Using the compound-angle identity, we have
x' = r cos a cos 0 + r sin a sin 6 ;
and, as x = r cos a and y = r sin a,
x' = x cos 0 + y sin 0.
Similarly,
sin (a - 6) = —
r
y' = r sin (a- 6);
= r sin a cos 0 - r cos a sin 0;
and, as x = r cos a and y = r sin a,
y ' = - x sin 0 + y cos 0.
Expressing Original Coordinates in New Coordinates
After coordinate rotation with the origin fixed, the original
coordinates (x,y) expressed with reference to the new
coordinates (x\y1) are
x = x' cos 6 -y' sin 0 and y = x' sin 0 + y' cos 0

Chapter 15 ANALYTIC GEOMETRY
To derive the formulas that express the original coordinates in
the new coordinates, we solve for x and y in the known
formulas:
(1) xl = x cos 0 + y sin 0
(2) y1 = -x sin 0 + y cos 0
Multiply (1) by cos 0, and (2) by sin 0; thus,
(1) => (3) x' cos 0 = x cos2 6 + y sin 0cos 0
(2) => (4) y' sin 0 = -x sin2 0 + y sin 0cos 0
Subtract (4) from (3); thus,
x' cos 0 - y' sin 0 = x cos2 0 + y sin 0 cos 0 - (-x sin2 0 + y sin 0 cos 0)
= x cos2 0 + y sin 0 cos 0 + x sin2 0 - y sin 0 cos 0
= x (cos2 0 + sin2 0),
508 simplified to
x = x' cos 0 - y' sin 0.
Similarly, multiply (1) by sin0, and (2) by cos0; thus,
(1) => (5) x' sin 0 = x cos 0 sin 0 + y sin2 0
(2) => (6) y' cos 0 = -x sin 0 cos 0 + y cos2 0
Add (5) and (6):
x' sin 0 + y' cos 0 = x cos 0 sin 0 + y sin2 0 -x sin 0 cos 0 + y cos2 0
= y (cos2 0 + sin2 0);
y = x' sin 0 + y 'cos 0 .
• A truss is a structural load-carrying system made from
geometrical arrangement of straight structural elements
which are connected at their ends only. The figure shows a
3-member truss.
When a specified load is applied to connection C (node C), C is
displaced vertically to C". With the displacement CC the truss
assumes a new configuration which is shown exaggerated by
dashed lines:
From precision theodolite measurement we know that the
displacement CC is 4.53 mm.
Determine the components of CC in x',y'-coordinates where
x' is taken through the axis of member BC

Section 15.3 Shifting Orthogonal Coordinates
575
The basic relationships are
x' = x cos 6 + y sin 6
y' = - x sin 6 + y cos 6.
From the measurement we have x = 0, y = 4.53 mm and from
the geometry sin 6 = 0.8 and cos 6 = 0.6 (complementary
angles); thus,
x1 = 0 + 4.53(0.8) ** 3.62 mm
yx = 0 + 4.53(0.6) & 2.72 mm.
Note: The 3.62-mm elongation of member BC is produced by
the stress occurring in BC as a consequence of the external
load applied at C, which in its turn causes the displacement
CC\ Since the deformations are small in relation to the
overall dimensions of the truss, it is customary to carry out
a first-order analysis, that is, to use the original geometry
even for the deformed configuration of the truss. Thus, 6
remains unchanged.

576
Chapter 15 ANALYTIC GEOMETRY
15.4 Polar Coordinate Systems
Radius Vector
Pole
Polar Angle, Vector
Polar Axis
In previous discussions the position of points has been
specified by means of orthogonal coordinates - ordered real
numbers in a plane (x,y) or in space (x,y,z). In some
situations, however, it is more convenient to locate a point by
means of polar coordinates, that is, a system of coordinates
where the position of a point is determined by
o the length of the ray segment (the radius vector) from a
fixed origin (the pole), that is, the distance between the point
and the pole, and
o the angle (the polar angle) the ray (the vector) makes with
a fixed line (the polar axis):
£V
SA
ice
\pS
pole +
3>^\polar angle
polar axis
The pole is usually denoted O; the length of the radius vector,
r or p; the polar angle, 6 or 0:
P = (r, 6)
O
Vectorial Angle, Argument,
Amplitude, Azimuth
Arabic, as-samut, "points of
the horizon"
Thus, in a polar coordinate system the position of a point is
determined by distance (r) and direction (0).
The polar angle is sometimes called the vectorial angle, the
argument, the amplitude, or the azimuth of the point.
Polar Coordinates and the Plane
In polar coordinates, each point P in the plane is specified
by two coordinates (r, 0), where r is the distance OP, and
- as in trigonometry - positive 0 is the angle measured
counterclockwise.
Contrary to the coordinates of a point in an orthogonal
coordinate system, polar coordinates are not unique; (4, 50°),
(4, 410°), and (4, -310°) all represent the same point.

Section 15.4 Polar Coordinate Systems
577
Polar Grid
A polar grid is formed by concentric circles and by rays with
endpoints at the center of the circles and at P. The first
(innermost) ring is of radius 1, the second of radius 2, and so on.
In the graph below, angles are expressed in radians:
5tc/6
P2(3, 471/3)
Pi (4, Ti/6)
Efe^orad
7tc/6
axis
Htc/6
3tc/2
Shifting between Orthogonal and Polar
Coordinates in the Plane
To convert between orthogonal and polar coordinates, let the
pole coincide with the origin of the orthogonal system and the
polar axis coincide with the jc-axis:
P =
(x, y) Cartesian
(r, 0) polar
polar axis
The graph enables us to see that the conversions between
orthogonal and polar coordinates are
x
y
r
r cos 6
r sin 6
>/x2 + y2 (the Pythagorean theorem)
6 = arctan —, if P is in the first or fourth quadrant.

580
Chapter 15 ANALYTIC GEOMETRY
K
Hyperbolic Spiral
In the polar equation
r 6 = a,
where a is a constant of proportionality, the radius vector r
varies inversely with the polar angle 0, so that the curve forms
a hyperbolic spiral. The spiral is asymptotic to a straight line
that is parallel to the polar axis and located at the distance a
above the axis:
n/2
Orad
► Orad
3ic/2
r6=a;0>O
3*/2
rO=a;0<O
VARIGNON Pierre
(1654 -1722)
The hyperbolic spiral was first conceived in 1704 by the French
mathematician Pierre Varignon.
Logarithmic Spiral
A polar equation written either in logarithmic form,
log6r = a 6,
where a is a constant of proportionality > 0, or in exponential
r = bae,
form,
represents a curve that is referred to as a logarithmic spiral or
an equiangular spiral:
CO
CD
-a t
£ s
<a*
8 fl
§&*
o « g
£ £ -S
»§ S
« o c
^o<3
71
► Orad
3tc/2
logfcr = a 0 r = ba6
The chambered nautilus does not grow in all directions but only
adds on to its open end, thereby maintaining its logarithmic spiral.
A notable property of the logarithmic spiral is that it intersects
its radii everywhere at the same angle - thus the name
equiangular spiral.

Section 15.4 Polar Coordinate Systems
581
DESCARTES Rene
(1596 -1650)
BERNOULLI Jakob
(1654-1705)
The logarithmic spiral was first discussed by Descartes in
1638; in 1698 its properties were further studied by the Swiss
mathematician Jakob Bernoulli, who was particularly
intrigued by its self-similarity: if any portion is magnified or
reduced, it is identical to any other portion of the curve.
Latin, lituus, "a crooked staff
COTES Roger
(1682 -1716)
The Lituus
The lituus (plural: litui) is asymptotic to the polar axis, and
winds around and gets increasingly closer to the pole but
never reaches it.
n
polar w
-—: ► 0 rad
Sn/2
The lituus originated with the English mathematician Roger
Cotes; it was published posthumously in a collection'of Cotes's
mathematical papers, Harmonia mensurarum (1722).
deFERMAT Pierre
(1601 -1665)
Parabolic Spiral (Fermat's Spiral)
In the parabolic spiral the square of the radius vector is
proportional to the vector angle. First discussed by the French
mathematician Pierre de Fermat in 1636, this spiral is also
referred to as Fermat's spiral.
7T/2
r2 = a0
Orad
Sn/2
The Lemniscate
Latin, lemniscatus, "with hang- The lemniscate, or lemniscate of Bernoulli, was conceived by
ing ribbons"; lemniscus, "ribbon" Jakob Bernoulli in an article on tides (1694).
n/2
r2 =a2 cos 20
polar ^
——: ► 0 rad
axis
Sn/2
Polar form: r2 = a2 cos 2 0 Cartesian form: (x2 + y2)2 = a2(x2-y2)

582
Chapter 15 ANALYTIC GEOMETRY
Polar Coordinates and Space
Cylindrical Coordinates
A system of cylindrical coordinates combines the use of polar
coordinates in the plane with the z-coordinate of three-
dimensional orthogonal coordinates.
The location of a point P (r, 0, z) is its projection (r, 0) onto the
x, y-plane and its distance z from the x, y-plane:
P(r, e, z)
(r,0)
x
Cylindrical Coordinates and Their Graphs
Axis of Symmetry
Cylindrical coordinates are useful primarily to reproduce
figures that have an axis of symmetry, which may be
conveniently placed at the z-axis of the graph.
o If 0 and z vary with r constant, a right circular cylinder
develops.
o If r and z vary with 0 constant, a half-plane with one side at
the z-axis develops.
o If z is constant and r and 6 vary, a half-plane with one side
at they-axis develops.
In the graphs below, c denotes a constant:

Section 15.4 Polar Coordinate Systems
583
Conversions between Orthogonal and
Cylindrical Coordinates
The equations connecting orthogonal and polar coordinates in
the plane also apply in space. The ^-coordinates of the
orthogonal coordinate system and the cylindrical coordinate system
coincide. We obtain the equations:
x = r cos 6
y = r sin 6
z = z
= v#2 + y2
6 = arctan —, if (x, y) is in the first or fourth quadrant.
Spherical Coordinates
In a system of spherical coordinates a point P (p, 0, <p) is located
by the distance of P from a fixed origin or pole O (distance p)
and two angles, denoted 6 and <f>. Angle 6 is the same as in
polar coordinates in the plane; <f> is the angle between the
positive z-axis and the line segment OP:
PiP, « ¢)
(r, e)
x
Center of Symmetry
Spherical Coordinates and Their Graphs
Spherical coordinates are useful primarily to reproduce
figures that have a center of symmetry, which may be
conveniently placed at the pole.
o If 6 and <j> vary with p constant, a sphere develops.
o If p and <f> vary with 6 constant, a half-plane with one side at
the z-axis develops.
o If p and 6 vary with 0 constant, a nappe of a right circular
cone develops.

584
Chapter 15 ANALYTIC GEOMETRY
In the graphs below, c denotes a constant:
x
x
Conversions between Spherical and
Orthogonal Coordinates
Why?
Formulas Transforming Spherical Coordinates into
Orthogonal Coordinates
To transform spherical coordinates into orthogonal
coordinates, we employ the formulas
x = p sin <f> cos 6
y .= p sin <f> sin 6
z = p COS <f)
(P.O.*)
(x, y, z)
(r,0)
x
Solving the triangle marked I, we find
sin <b = —
P
(1)
and that
r = p sin <f>
cos ¢ = -
Y P
z = p cos <S>.

Section 15.4 Polar Coordinate Systems
585
Using Equation (1) when solving the triangle marked II, we
also find
and, finally,
cos 6 =
x
p sin <j>
x = p sin (f> cos 6
sin 6 =
y
p sin 0
y = p sin 0 sin 6.
Formulas Transforming Orthogonal Coordinates
into Spherical Coordinates
To transform orthogonal coordinates into spherical
coordinates, we employ the formulas
p = -\lx2 +y2 + z2
6 = arctan —, if (x, y) is in the first or fourth quadrant
Why?
x
0 = arccos
-\[x2 +y2 + z2
fe y, z)
(r,0)
Using the formula for distance in space and referring to the
above graph, we find
p = Vx2 +y2 + z2 ,
and from the triangle marked II,
tan 6 = — ;
x
from the triangle marked I,
z
cos (j> =
V x2 +y2 + z2
; (/> = arccos
V x2 +y2 + z2

Chapter 15 ANALYTIC GEOMETRY
15.5 Parametric Equations
Instead of representing a two-dimensional graph by one
equation in two variables, it may be represented by two equations,
each of which gives the coordinates of x and y in terms of a
third variable, called a parameter and usually denoted t.
Equations with parameters are called parametric equations;
they often facilitate the handling of complex graphs.
Consider the equation
y2 = 4 x .
If we let y = 2t, then we have the parametric equations
x = t2 and y = 2t,
representing the parabola.
Similarly, a three-dimensional graph may be represented by
three parametric equations, each of which gives the
coordinates of x, y, and z in terms of the parameter.
There is no general method of transforming an equation
representing an orthogonal or polar graph into parametric
form; such transformation often takes considerable
persistence and ingenuity.
The Circle
The parametric representation of a circle with radius r and
centered at the origin is
x = r cos t
>
y = r sin t
Why? Consider the equation x2 + y2 = r2, that is, a circle with radius r
and its center at the origin of an orthogonal coordinate system:
y
i
r
L
^:
j
x . y
cos t = — sin t = —
r r
x = r cos t y = r sin t

Section 15.5 Parametric Equations
587
The procedure for finding the parametric representation of a
circle centered outside the origin of an orthogonal coordinate
system follows the same course of reasoning as for a circle
centered at the origin.
Express x and y of
(x - 4)2 + (y - 3)2 = 4
as parametric equations in t, where t is the central angle of the
circle.
The equation represents a circle whose radius is "v4 = 2 and
whose center in the orthogonal coordinate system is at (4, 3):
x
cos t =
X
-4
x = 4 + 2 cos t
sin t =
y-3
y = 3 + 2 sin t
The sought parametric equations are
x = 4 + 2cos£
y = 3 + 2 sin t
The Ellipse
► x
The parametric representation of an ellipse centered at the
origin is
x = a cos t
y = b sin t
where 2a and 26 are axes of the ellipse, such that if a > 6, the
foci of the ellipse are on the x-axis, and if 6 > a, the foci are on
the y-axis; if a = 6, then the curve is a circle.
To find a parametric representation of an ellipse, use
auxiliary concentric circles and let an ellipse whose major
axis is 2 a and minor axis is 2 6 be placed with its center at the
origin of an orthogonal coordinate system, and let t be the
angle that an arbitrary ray from the origin makes with the
x-axis; then
x = a cos t
y = b sin t

588
Chapter 15 ANALYTIC GEOMETRY
The Hyperbola
A hyperbola centered at the origin and with the transverse axis
2a (coinciding with the x-axis of an orthogonal coordinate
system) and conjugate axis 26 has the parametric equation
x = a sec t
y = b tan t ,
To deduce this system of equations, consider two concentric
circles centered at the origin and with diameters 2 a and 2 6,
equal to the transverse axis and the conjugate axis of the
hyperbola, respectively:
y
A
'* x i tC y^^ «>
b
i
\i/ [ \ *
\ i transverse N
I 1 0'
l\ I >•■**
M . \ jC "O
/ \s\ c
/ i /\ \^ o
X 1^ x. ^"-"- ^
- tt -1
^N^/x
/
v^ \ \l/
S axis |
n. J /i\
nV /i\
/Y i \
i_—^^^ / N.I X
X r
The x-axis intersects the inner circle at a point lif.
Let the x-axis and an arbitrary ray from the origin form an
angle t.
A tangent to the inner circle at K intersects the arbitrary
ray at L; the ray intersects the outer circle at M.
The tangent to the outer circle at M intersects the x-axis
atiV.
Lines parallel to the x-axis and y-axis and passing through
L and N, respectively, intersect at a point P (x, y) on the
hyperbola.
P(x, y)
x

Section 15.5 Parametric Equations
589
From the right triangle OMN, we find
x
sec t = —
a
x = a sec t.
Similarly, from the right triangle OLK, we find
y
tan£ = 7-
b
y = b tant.
The Parabola
p. 586
In the introduction to this section, we found that a parabola
whose orthogonal equation is y2 = 4x may be represented by the
parametric equations
x = t2
y = 2t
which represent a parabola whose y-axis is a tangent at the
conjoining points of the origin of the orthogonal coordinate
system and the vertex of the parabola:
x
Path of a Projectile
Verification: pp. 901 - 902
Different forms of parametric equations may be derived to suit
different types of problems. An example is the parametric
equations of the parabola, used to determine the path of a
projectile, here quoted without proof; thus,
x = v0 t cos 6
y = votsm0--rgt2
>
where i;0 is the initial velocity of the projectile, 0the initial
angle between the projectile and the horizontal plane, g the
acceleration of gravity, and t the time elapsed.

590
Chapter 15 ANALYTIC GEOMETRY
Cycloids
Cusp
Latin, cuspis, "a point"
A cycloid is the curve generated by a point on the
circumference of a circle which rolls on a straight line in its plane.
At a completed revolution of the circle and the beginning of the
following cycle a double point - a cusp - is formed:
cusp
WREN Christopher
(1632 -1723)
GALILEI Galileo
(In English literature referred
to by first name, "Galileo";
1564 -1642)
ROBERVAL Gilles Personne de
(1602 -1675)
Properties of the Cycloid
The length of the cycloid baseline - the distance between
successive cusps - is equal to the circumference of the
generating circle.
The length of a cycloid between successive cusps is four times
the diameter of the generating circle. Alternatively, the
length of one complete arc of a cycloid may also be
determined by multiplying the corresponding distance along the
baseline by 4 and dividing the product by n. For instance, if
the baseline distance between two cusps is Z, then the length of
the arc is 41/n.
In 1658 the English architect and mathematician Christopher
Wren proved that the length of one complete arc of a cycloid
equals the perimeter of a square circumscribed about the
generating circle. This agrees with the results above.
By the use of models and comparing the weights of the circle
and the cycloid it generates, Galileo demonstrated in 1599 that
one complete cycloidal arch is about three times the generating
circle in area; in 1634 the French mathematician Roberval
proved that the area of one complete cycloidal arch is indeed
three times that of the generating circle.
BERNOULLI Johann
(1667-1748)
Greek, brachys, "short";
chronos, "time"
The Path of Most Rapid Descent
In 1696 the Swiss mathematician Johann Bernoulli
challenged fellow mathematicians to find the curve along which
a particle will slide - under the influence of gravity - in
the shortest time between two points at different altitudes but
not on the same vertical line, a problem referred to as the
brachistochrone problem.
Johann Bernoulli had found the solution; among those who
picked up the gauntlet and solved the brachistochrone problem
were Newton, Leibniz, L'Hospital, and Johann's brother Jakob
Bernoulli.

CHRISTIANI
H V G E NI I
ZVLICHEMII. CONST F
HOROLOGIVM
OSCILLATORIVM
SI VE
DE MOTV PENDVLORVM
AD HOROLOOIA APTATO
DEJtIOHSTlATIOHES
QtOMITlICA
ifU f. MttttT, fctgii * Uk&rifiai AJckkpfeft Trfefi*rh«.
m»CiikMm,Ui*CpwMit«i«».
MOCLXXIIL
CFM TktrtLEClQ XICIS.
HUYGENS Christiaan
(1629 -1695)
Greek, isos, "equal";
chronos, "time"
®
0
Section 15.5 Parametric Equations
591
One might feel inclined to guess that the shortest distance
would also be the quickest, but the answer is that the quickest
path is along a cycloid.
0
©
Path along a straight line
Path along a cycloid
In Horologium Oscillatorium (1673) - a work of fundamental
importance for the development of dynamics - the Dutch
mathematician, astronomer, and physicist Christiaan Huygens
described the isochronous property of the cycloid, that is, the
property that wherever a particle is placed on the concave side
of a cycloid, the time for sliding down to the lowest point of the
cycloid is - disregarding friction - always the same:
9
Pendulum Property
Another famed quality of the cycloid is its pendulum property:
If a pendulum is hung at the cusp of an inverted cycloid and
made to wrap around the arcs of the cycloid as it swings, the
end of the pendulum describes another cycloid and the period
of oscillation of the pendulum is independent of the amplitude;
as shown by Huygens, the period of the pendulum becomes
dependent only on its length and its shape.
Parametric Equations of the Cycloid
A cycloid whose generating circle has the radius a has the
parametric equations
x = at -a sint
y = a —a cos t
To deduce the equations, let the circle roll on the x-axis of an
orthogonal coordinate system:
x
(27ca, 0)

592
Chapter 15 ANALYTIC GEOMETRY
Use the angle of rotation (t) as a parameter to express the curve
of the cycloid:
x
(2rca, 0)
By examining the above graph, we find
x = at - a sint
y = a — a cos t
Epicycloids
An epicycloid is a curve generated by a point on the
circumference of a circle (the epicycle) which rolls on the outside of a
fixed circle (the deferent). More complicated curves can be
generated by having a 3rd circle roll on the 2nd, a 4th on the
3rd, etc. Epicycloids were introduced by the ancient Greeks
and, until the 16th century, formed the basis for ideas about the
paths of the Moon and planets. They still have significance in
engineering and present interesting mathematical properties.
Parametric Equations of the Epicycloid
An epicycloid whose generating fixed and rolling circles have
the radii a and b, respectively, has the parametric equations,
here quoted without proof,
i a + b x ^
x = (a + b) cos t - b cos I —:— t
y = (a + b) sin t-b sin I —-— t
>
j
where a is the radius of the fixed circle and b the radius of the
circle rolling on the fixed circle. The parameter t is the angle
formed by the x-axis and the ray from the center of the fixed
circle to the point of contact with the rolling circle:
(x, y)
x

Section 15.5 Parametric Equations
593
The epicycloid has a cusp at every point where it meets the
fixed circle.
The epicycloid describes one arc when the radius of the fixed
circle and the radius of the rolling circle are equal (a = 6), two
arcs when a = 26, and n arcs when a = nb.
Hypocycloids
A hypocycloid is a curve generated by a point on the
circumference of a circle which rolls on the inside of a fixed circle.
Parametric Equations of the Hypocycloid
A hypocycloid whose generating fixed and rolling circles
have the radii a and 6, respectively, has the parametric
equations, here quoted without proof,
/ ,x , .a -b
x = (a - 6) cos t + b cos I —;— t
y = (a -b) sin t-b sin I —;— t
>
The parameter t is the angle formed by the jc-axis and the ray
from the center of the fixed circle to the point of contact with the
rolling circle:
(*, y)
x
Like the epicycloid, the hypocycloid has a cusp at every point
where it meets the fixed circle; it describes two arcs when the
radius of the fixed circle is twice the length of the radius of the
rolling circle (a = 26), and n arcs when a -nb.
* * *
Let the bottom coin roll half way around the fi?(ed upper one.
- Which way will George Washington's head point?
Answer: '(Wod Sui^ib^s 8i# ye re) pjBMdn

594
Chapter 15 ANALYTIC GEOMETRY
The Astroid
A hypocycloid whose rolling circle has a diameter one-fourth
of that of the fixed circle is called an astroid.
(x, y)
x
By applying the parametric equations of the hypocycloid to the
astroid and then simplifying the equations, we come to the
result that the parametric equations of the astroid are
x = a cos31
y = a sin31
where a is the radius of the fixed circle.
Using the general parametric equations of the hypocycloid,
derive the formula for the astroid; a and 6 are the radii of the
fixed and rolling circles, respectively.
.-«»-».-.+»«r±^.|.»a.-.«.s«)
'a a a
y - (4 6 -b) sin t-b sin | —z— t ] = 6 (3 sin t- sin 3 t)
>
p. 511
As
cos 3 £ = 4 cos3 £ - 3 cos t; sin 3 £ = 3 sin t - 4 sin31 ,
we find
x = b (3 cos t + cos 31) = 4 6 cos31
y = b (3 sin t - sin 3 t) = 4 6 sin31 J ,
and since a = 4 6 ,
x = a cos31
y = a sin31

Section 15.5 Parametric Equations
595
Notable Three-Dimensional Spirals
The Cylindrical Helix
Greek, helix, "a spiral", "a twist"
APOLLONIOS
(c. 255 -170 B.C.)
p. 408
p. 586
The cylindrical helix - often simply called a helix - is a
three-dimensional curve formed as if lying on a right
circular cylinder, where it cuts the generators of the surface at
a constant angle a.
The helix is mentioned in a text by Geminus dating from the
1st century B.C., but a passage in the text suggests that the helix
was already known to the Greek mathematician Apollonius,
author of Conies.
Familiar structures of helix-like form are the threads of bolts
and lightbulbs, climbing plants, the human umbilical chord,
and the the famed double helix of DNA.
The parametric representation of the cylindrical helix is
x = a sin t ^
y = a cos t
z = bt
J
where a is the radius of the cylinder, 6 is a constant, and t is the
parameter, obtained as for the circle.
As t increases from 0 to 2n,
the height z steadily increases
from 0 to 2 n b, completing one
full turn of the helix.

596
Chapter 15 ANALYTIC GEOMETRY
The Conical Helix
The conical helix is a three-dimensional curve formed as if
lying on a right circular cone, where it cuts the generators of
the surface at a constant angle a.
The parametric representation of the conical helix is here
quoted without proof:
x = bet sint
y = be* cost
\
z = e*
j
where 6 is a constant and t is the parameter.
The Spherical Helix
Greek, loxos, "oblique";
dromos, "course"
NUNEZ SALACIENCE Pedro
(In English often referred to
as Peter Nunes; 1502 -1578)
MERCATOR Geradus
(1512-1594)
160o120°80°40" 0'
80° 120°160
:=1-J 70'
Mercator projection
reat Circle
(Orthodrome)
-A Equator
The spherical helix - better known as a loxodromic spiral, or
just loxodrome, or a rhumb line - cuts the meridians of a
sphere at a constant angle a, not equal to a right angle. The
loxodrome coils around the poles of the Earth, but never
reaches the poles.
The shortest distance of traveling between two points on a
sphere is along the arc of a great circle (the orthodrome) and to
stay the course, one must every instant adjust to the direction
shown by the compass. Along the loxodrome, the path to the
destination is longer, but it is always directed to the same point
of the compass.
The loxodrome was invented by the Portuguese
mathematician and geographer Pedro Nunez after observations reported
to him in 1533 by Admiral Martim Alfonso de Sousa.
In 1569 the Flemish cartographer Geradus Mercator designed
a world map with a cylindrical projection such that meridians
and parallels were straight lines, intersecting at right angles,
and whose distance from each other increased with their
distance from the equator. The Mercator projection distorts the
image (especially at high latitudes) but has the great
advantage of showing loxodromes as straight lines; such maps are
still important tools for navigation at sea and in the air.
The loxodrome can be written in terms of the longitude and
latitude of a point on the curve and its angle with the
meridians; there is no advantage in using parametric equations.

Section 15.5 Parametric Equations
597
Eliminating the Parameter
■*- *"~r*v/*JS^
A real bear... eliminating
the parameter.
To eliminate a parameter from a parametric equation and
thereby obtain an orthogonal or polar equation is often at least
as complex as forming a parametric equation.
Find the corresponding orthogonal equation of the parametric
equations
x = 7 + 3 cos t
>
y = 4 + 3 sin t
x-7 . . y-4
cos t = —-— , sin t = —-—
Pythagorean Identities: p. 508
* - 7^2+ (y: 4Y = x
(x - 7)2 + (y - 4)2 = 9
which represents a circle with radius 3 and center at point
(7, 4) in an orthogonal coordinate system.
Transform the parametric equations
Pythagorean Identities: p. 508
x
y
z
V5 suit
3cos£
>
= t
j
into a general equation, relating x andy.
x v
We note that -=■ = sin t and — = cos t, so that
V5 3
x2 y2
T+9"=1>
which is the equation of a right elliptical cylinder:
^
^_
^_ t
->
^J
■»
2^5~
2k
y

598
Chapter 15 ANALYTIC GEOMETRY
Find the corresponding general equation of a parabola
represented by the parametric equations
(1)
(2)
Solve (1) for t:
Substitute (x — 3) for t
x = t + 3
J
>
v — JC o .
in (2):
(x - 3)2
(3) (*-3)2 = 4y,
which is the sought equation, representing a parabola with the
axis parallel to they-axis, vertex at (3, 0), and focus at (3, 1).
* * *
I say, (Doctor, what about making the cut y = 2x + 7; -5^^^1^

599
Chapter
16
VECTOR ANALYSIS
Page
16.0 Scope and History 600
16.1 Basic Vector Algebra 602
16.2 Scalar and Vector Components 607
16.3 Multiplication of Vectors 614
*** 24 Complex Numbers Revisited 787
*** 30.4 Velocity Vectors and Acceleration Vectors 901
*** Indicates cross-reference

600
Chapter 16 VECTOR ANALYSIS
16.0 Scope and History
STEVIN Simon
(1548-1620)
Vectors are directed line segments that have both magnitude
and direction. They are used to describe physical quantities
like velocity and force, and are usually symbolized by arrows.
The method of calculating the combined action of two or more
mechanical forces by adding them according to the
parallelogram law had been known ever since the time of Aristotle
when, in 1586, the Flemish mathematician Simon Stevin
analyzed the principle of geometric addition of forces in his
treatise De Beghinselen der Weeghconst ("Principles of the
Art of Weighing"), which caused a major breakthrough in the
development of mechanics. But it would take another 200
years for the general concept of vectors to form.
D £
BEGHIKSELEK
DEH WEEGHCONST
BESCHREVEN OVER
% iU O H STIYttf
van Brugghc
A chain around a triangular support,
as shown here, was at one time
thought to be, through the influence
of gravitation, capable of perpetual
motion. Stevin showed that the
device does not work as a perpetual
motion machine.
In translation, the inscription reads:
"To wonder is no wonder/'
T«t Ltrai*,
tndc Drockoyx m Ouiftml Ptioajn,
WESSEL Caspar
(1745-1818)
ARGAND Jean Robert
(1768-1822)
WARREN John
(1796-1852)
HAMILTON William Rowan
(1788-1856)
GAUSS Carl Friedrich
(1777 -1855)
The germinal ideas of modern vector theory date from around
1800 when Caspar Wessel and Jean Robert Argand described
how a complex number a + b i could be given a geometric
interpretation in a coordinate plane. Based on the work of Argand,
the English mathematician John Warren in 1828 published
A Treatise on the Geometrical Representation of the Square
Roots of Negative Numbers, which inspired the Irish physicist-
astronomer-mathematician W. R. Hamilton to show, in 1837,
that complex numbers can be regarded as ordered pairs of real
numbers. Unknown to Hamilton, German mathematician
Carl Friedrich Gauss had made the same discovery six years
earlier, in 1831.

Section 16.0 Scope and History
601
Quaternion
Latin, vectus, perfect participle of
vehere, "to carry"
vector, "one who carries"
GRASSMANN Herman Gunther
(1809 -1877)
GIBBS Josiah Willard
(1839 -1903)
Continuing his investigations, Hamilton found that a hyper-
complex number required an ordered set of four real numbers
for its representation; a + 6i + cj+dk, which Hamilton called
a quaternion, would do if one sacrificed the commutative law
of multiplication of traditional algebra:
ij = -ji = k; jk = -kj = i; ki = -lk = j
and i2 = j2 = k2 = ijk = _x
Just as there are both Euclidean and non-Euclidean
geometries, there are also several kinds of algebras that, within
their respective realms of validity, are as free of
contradictions as conventional algebra.
Hamilton was the first to use the term vector for a directed line
segment in his book Lectures on Quaternions (1853), which
overflowed in peculiar terms: besides vector, there were
vehend, vection, vectum; revector, revehend, revection,
revectum; provector ...; transvector ...; etc., which made the
book all but unintelligible to the majority of contemporary
mathematicians.
In 1844, the German mathematician Hermann Grassmann -
independent of Hamilton - published Die lineale Ausdeh-
nungslehre ("Theory of Linear Extension"), in which he
treated n -dimensional geometry and hypercomplex systems
in a much more general way than did Hamilton.
From Grassmann's and Hamilton's ideas, American
mathematician and physicist J. W. Gibbs developed much of vector
analysis as we know it today; his treatise Vector Analysis
appeared in 1881. Yet quaternions in their original form are
not forgotten but have an established place in algebra and in
quantum physics.
* * *
fMany things have more than direction;
7¾ magnitude is also a question.
With acceleration or force,
And many more things, of course,
It's vectors that make the connection.

602
Chapter 16 VECTOR ANALYSIS
16.1 Basic Vector Algebra
Scalars and Vectors
Latin, scala, "a ladder"
Latin, vectus, "carried"
terminal
pointy o
^ initial
point
Quantities such as mass, length, volume, pressure,
temperature, voltage, and time can be characterized by a single real
number, scaled to a suitable unit of measurement. Such
quantities are called scalar quantities, and the real number
associated with the magnitude (value) of the quantity is called
a scalar.
On the other hand, quantities such as force, velocity, and
acceleration require, for their definition, the assignment of
not only a magnitude but also a direction. Such quantities are
called vector quantities and may be represented
mathematically by two or more real numbers.
Geometrically, a vector is represented by an arrow (a line
segment) with a specific direction, its length corresponding to
the magnitude of the vector. A vector representing a
displacement from a point A to a point B is said to have the initial point
or origin A and the terminal point or terminus B.
Several notations are currently in use for vectors and for the
magnitude of vectors. In this text we use the following
notations:
a
a | or a
0
which reads
"Vector a"
"Magnitude of vector a"
"Null vector"
In handwritten or typewritten text it is customary to underline
letters or to use overbars to indicate vector character,
a or a and OorO
—>
In both printed and handwritten texts the notation AB is used
for a vector displacement from A to B.
Vectors are equal if they have the same direction and the same
magnitude. Of the vectors below, a equals b, and c equals e:
A scalar multiple is the product of a scalar and a vector to
produce another vector.

Section 16.1 Basic Vector Algebra
603
Null Vector
If a is a vector, then the scalar multiple /a is a vector with
o the same direction as a if y > 0;
o the opposite direction of a if y< 0;
and
o |/a| = \y\ | a|.
Below, vector a is multiplied by the scalars 0.5, 1.5, and -1:
A vector a equals zero if its magnitude | a | is zero; it is called
a null vector or zero vector, and is denoted 0.
The null vector is the only vector whose direction is
indeterminate.
Resultant
Adding Vectors
A vector that is the sum of a given set of vectors is called the
resultant.
Vectors in Line
Addition of vectors in one dimension (vectors on one line) is
similar to arithmetical addition. Geometrically, the vectors
are positioned along a straight line so that the initial point of
one coincides with the terminal point of the other:
-^+
^4
Triangle and Parallelogram Laws
To add vectors in two dimensions geometrically, we position
them so that the initial point of one and the terminal point of
another coincide ("head to tail").
+~Q
To add vector b to vector a, we shift b, without changing its
direction or magnitude, so that its initial point coincides with
the terminal point of a.

604
Chapter 16 VECTOR ANALYSIS
Triangle Law of Addition
Parallelogram Law of Addition
Thus, according to the triangle law of addition the sum of a
and b equals the displacement of P to R:
Addition of
is commutative.
—>
a + b= PR
a+b=b+a
To study the additive property of vectors a and b,
we displace b so that its initial point coincides with the
terminal point of a; the terminal point of b will then fall at Q;
similarly, if a is displaced to make its initial point coincide
with the terminal point of b, the terminal point of a will fall
atQ:
Q
P a
We thus have the commutative property of addition,
a+b = b + a;
and
a + 0 = a; a+ (-a) = 0.
The above figure illustrates the parallelogram law of addition,
another presentation of the triangle law of addition.
The associative property of addition of numbers says that
a + (b + c) = (a + b) + c .
To show that vectors also satisfy this property, we arrange
vectors a, b, and c as the consecutive sides of a polygon:
S
Applying the triangle law of addition, we see that the
displacement of point P to point R equals the sum of a + b, and the
displacement of Q to S equals the sum of b + c.

Section 16.1 Basic Vector Algebra
605
Consequently, by the associative property of addition:
a + (b + c) = (a + b) + c
As with addition of scalars, we denote the two equal sums by
a + b + c.
R
a / >^
^/ a + b + c x.
pi ^g
The resultant of a larger number of vectors is found by
positioning the vectors, successively, "head to tail". The resultant
is the straight line joining the initial point of the first vector
and the terminal point of the last vector:
The following example refers to force (F) vectors. The SI unit
newton (N) of force is the newton (N). 1 N is that force which, when applied
to a body having a mass of 1 kg, gives it an acceleration of
1 m/s2.
• A freight car is being pulled by two cables as shown in the
figure below.
(2)
Iur_
12°
(1)
The resultant of the tensile forces has a magnitude of 33.50
kilonewtons.
Find the magnitudes of the tensions in the cables (1) and (2)
assuming no side force from the rails.
The freight car moves in the direction of the rails, and the
resultant must lie along that axis. Let the tensile forces in
cables (1) and (2) be Ti and T2, respectively. Construct the
force polygon:
t

Chapter 16 VECTOR ANALYSIS
By the law of sines we have:
33.50 kN 33.50 kN Tx T2
sin 158° sin 22° sin 10° sin 12°
Tx = 33.50 • Smoo0 * 15.53 kN
x sin 22
■ -I qO
T2 = 33.50 • . 000 * 18.59 kN
* sin 22
Hence, the tensile forces are 15.53 kN in cable (1) and 18.59 kN
in cable (2) .
Note: No consideration has been taken to the resultant
sideload on the rails.
Subtracting Vectors
The sum of two parallel vectors of equal length but opposite
directions is a null vector:
x = -a; a + x = 0
The difference of two vectors a and b is defined by
a - b = a + (- b).
Geometrically the difference between the two vectors a and b is
attained by translating the vector (- b) so as to make its initial
point coincide with the terminal point of vector a, and apply the
triangle law of addition.

607
16.2 Scalar and Vector Components
Right-Handed Coordinate Systems
In studying vectors, the use of an orthogonal coordinate system
is convenient. In this text the coordinates will be denoted x, y,
and z; in other texts the symbols x\> x^ and x% may be used.
It is customary in vector analysis to consider right-handed
systems only. When the positive x-axis is rotated
counterclockwise into the positive y-axis, a right-handed
(standard) screw would advance in the direction of the positive
z-axis.
In a right-handed coordinate system, the thumb of the right
hand points in the direction of the positive z-axis when the
fingers are curled in the direction away from the positive
x-axis toward the positive y-axis.
■+~y
y
X
Unit Vectors
A unit vector is a vector whose magnitude is 1.
Unit vectors in the direction of the orthogonal coordinate axes
are denoted ex, ey, ez when the coordinates are denoted x, y, z;
they are ei, e2, e3 for coordinates x\, X2, x%. The symbols i, j,
and k may be used irrespective of the notation of the
coordinates.
In a two-dimensional orthogonal coordinate system, vectors
i and j are the unit vectors from the origin to the points (1,0)
and (0, 1), respectively; in a three-dimensional system the

608
Chapter 16 VECTOR ANALYSIS
unit vectors i, j, and k are (1, 0, 0), (0, 1, 0), and (0, 0, 1),
respectively:
A
2--
lu
x
A
2--
The Position Vector and Its Components
Position Vector
Scalar Components
Vector Components
A vector represented by a line segment beginning at the origin
of an orthogonal coordinate system is called a position vector,
and is often denoted r.
A vector r with its initial point at the origin of an orthogonal
coordinate system and its terminal point at (x, y, z) can be
written
r = jci+yj +z k,
where x, y, z are the scalar components of r, and x i, y j, z k are
the vector components:
(x, y, z)
*~y
X
A vector r,
r = xi+yj+zk,
multiplied by the scalar A ,
Ar = X(x i +y j +z k) = (A#)i +(Xy)j +(Az)k,
has components A#, Ay, and Az.

Section 16.2 Scalar and Vector Components
609
p. 550
The magnitude of a space vector r may be calculated by the use
of the distance formula:
|r| = \x2 + y2 + z2
x
Given: vectors a = 4i-j-4k and b = -i + 3j+k
Find the vector sum and difference of a and b, and their
magnitudes | a + b | and | a - b |.
a + b = (4i-j-4k) + (-i + 3j+k) = 3i + 2j-3k;
|a + b| = ^32 + 22 + 32 = V22.
a-b = (4i-j-4k)-(-i + 3j + k) = 5i-4j-5k;
|a-b| = V52 + 42 + 52 = ^6.
The illustration below shows vectors a, b, a + b, and a-b. The
translated vectors b and -b are shown as dashed lines:

Chapter 16 VECTOR ANALYSIS
Three force vectors Fi, F2, and F3, with magnitudes 150, 210,
and 195 newtons, respectively, have the directions shown in the
figure below:
x
Find the magnitude R and direction 6 of the resultant R.
R = Fj + F2 + F3.
If we let Flx , F2x , F3x and Fly , F2y , F3y be the x and y
components of Fj, F2, and F3, and let Rx and Ry be the x and y
components of R, then
R = ^i+^j = (Flx + F2x + F3x)i + (F1 +F2 +FS )j.
F1y = 150 cos 30° *
F2 = 210 sin 20° *
F3 =-(195 cos 8°) ^-
^JC =
129.9 N
71.8 N
-193.1 N
8.6 N
Fi = 150sin30c
y
75.0 N
Zy
^3V =
-(210 cos 20°) * - 197.3 N
195 sin 8° * 27.1 N
#,
95.2 N
R = 8.6 iN - 95.2 jN
R = -\JS.62 + 95.22 * 95. 6
tan0 = ^ = 4P ; 6 =-84.8°.
rt<
8.6
The resultant is 95.6 newtons at - 84.8° .
According to Newton, an object moves in the direction of the
force acting on it. For the object to be stationary, the resultant
of all forces acting on it must be zero; that is, the resultant
must be a null vector. This state, when all forces acting on an
object are self-balancing, is a state of equilibrium.

Section 16.2 Scalar and Vector Components
611
-102.1 j N
63.2 i N ^ ^
Mass
Weight, Force
Acceleration of Free Fall (g)
The three vectors Fi, F2, and F3 of the preceding example pull
at an object. Determine how F2 must be changed in order to
keep the object motionless.
Fly * 75.0 N
FSy * 27.1 N
We have
Fi * 129.9 N
Fs ^-193.1 N
Equilibrium of the object requires that
Rx = 129.9 N +F2v - 193.1 N = 0; F2 = 63.2 N
Ry = 75.0 N +F2 + 27.1 N = 0; F2 =- 102.1 N
and
F2 = (63.2 i- 102.1 j)N.
The magnitude, F2, is
F2 = V(63.2)2 + (-102.1)2 * 120.1 N.
tan0 = -
102.1
63.2
0 = - 58.2(
Hence, to produce equilibrium, the vector F2 must be
120.1 newtons acting in the direction - 58.2°. •
In the following three-dimensional equilibrium problem
reference is made to the quantities mass and weight. The
mass (m) of a given body is independent of its location on
Earth or in the universe. The weight of a body is a force (F)
that equals the product of the mass of the body and the
acceleration of free fall (g). At sea level, g varies from 9.78
m/s2 at the equator to 9.83 m/s2 at the North and South Poles.
A structural unit is being lifted by means of three cables, AB,
AC, and AD. The unit has a mass of 78.5 kilograms (kg). The
origin of the coordinates (0, 0, 0) is taken at A, and the
coordinates of points B, C, and D are given below (m stands for
meter):
B (-0.12, -0.12,
C (-0.12, 0.32,
D ( 0.32, - 0.12,
g= 9.81 m/s2
Find the tensile force in newtons in each cable.
- 1.00) m
- 1.00) m
- 1.00) m
The main cable at A must carry the total weight (F) of the
structural unit:
F = 78.5 [kg] • 9.81 [m/s2] = 770 N

612
Chapter 16 VECTOR ANALYSIS
We construct a force diagram at A
F
AD
Express the forces F^b, F^c> Fad as vectors:
Fab = (FAB^eAB
FAC = (FAc)eAC
Fad = (^ad) *ad
>
j
where Fab is the magnitude of F^b > and &ab is a unit vector in
the direction of AB, etc.
The unit vector e^B is
*AB =
TAB (XB ~XA)i + (yB- yA) J + (ZB- za) k
rAB
V (xb - xA)2 + (yB- yA)2 + (zb - za)2
y kZltKs* *
where
tab = - 0.12 i-0.12j-1.00 k
YAC = - 0.12 i + 0.32 j - 1.00 k
tad = 0.32 i - 0.12 j - 1.00 k
and thus,
rAB
rAC
rAD
= V 1.0288
= V 1.1168
>
= V 1.1168
eAB -
eAC ~
e^n =
- 0.12 i - 0.12 j-1.00 k
V 1.0288
- 0.12 i + 0.32j-1.00 k
V 1.1168
0.32 i-0.12j-1.00 k
^
>
V 1.1168
and
^AB
^AC
F/in
= (Fab) :
= (Fac) :
= (Fa n)
-0.12 i-0.12j-1.00 k
V 1.0288
-0.12 i + 0.32j-1.00 k
V 1.1168
0.32 i-0.12j-1.00 k
>
V 1.1168

Section 16.2 Scalar and Vector Components
613
The equilibrium conditions are
(Fab)x + (FAc)x + (Fad)x = 0
(FAB)y + (Fac\ + (Fad\ = 0
(Fab)z + (Fac)z + (Fad)z = -770
Substituting the appropriate component values, we have
0.12 FAB 0.12 FAC 0.32 Fad
V 1.0288 V 1.1168 V 1.1168
0.12 FAB 0.32 FAC 0.12 Fad
V 1.0288 V 1.1168 V 1.1168
1.00 FAB 1.00 FAC 1.00 Fad
V 1.0288 V 1.1168 V 1.1168
= 0
= 0
= -770
^
>
J
Solving the above system of equations, we find
Fab = 355 N
FAC = 222 N
Fad = 222 N
^v
J

614
Chapter 16 VECTOR ANALYSIS
16.3 Multiplication of Vectors
Products of vectors can be formed in several ways; some of
these products are of special interest and importance because of
their geometrical interpretations and their uses in describing
various phenomena in physics.
We distinguish between scalar products and vector products.
The Scalar Product
A scalar product, a • b, is the sum of the products of
corresponding components of the vectors.
The scalar product is also known as the inner product, or the
dot product, and reads "a dot b".
If
then
a = xii+yj and b =x2i+y2J
ab = x\x2+y\y2.
The scalar product is useful for determining the angle between
two vectors.
The Scalar Product Theorem
Why?
p. 483
p. 550
The scalar product of two vectors is equal to the product of the
magnitudes of the two vectors and the cosine of their included
angle:
a b = I a I I b I cos 6
cos 6 =
a • b
al lb
The law of cosines gives
\PQ\2 = |a|2+ |b|2-2 |a| |b| cos0.
With
a = xii+yij and b = x2i+y2i
and
|PQ|2 = (x2-x1)2 + (y2-yi)2f
we obtain
(x2-x1)2 + (y2-yi)2 = |a|2+ |b|2-2 |a| |b| cos 6

Section 16.3 Multiplication of Vectors
615
Applying the Pythagorean theorem on triangles I and II, we
have
|a|2 = xi2 +yi2 and |b|2 = x<£ + y<£
and may write
(x2-x{)2 + (y2-yi)2 = (*i2 + yi2 ) + (x22 +^22)-2 lal |b| c°s 0
and
*i*2+yi:y2 = l»l |b| cos e.
We have already defined the scalar product
a b = xix2+y\y2,
and now have
and
a • b = | a | | b | cos 6
a b
al lb
cos 6 =
We note that the scalar product is a scalar and not another
vector. It is not to be confused with the scalar multiple A a of a
vector a.
For three-dimensional vectors we have, again:
a • b = I a I I b I cos 6
Scalar Products of Unit Vectors
Commutative Properties
Distributive Properties
Properties of Scalar Products
Since the cosine of 0 is 1, and the cosine of 90° is 0, the scalar
products of unit vectors are
and
i . i = j • j = k • k =11 cos 0 = 1
i-j = j.k = k i = 11 cos 90° = 0
Because the coordinates x\9 X2, yi, J2 are real numbers
satisfying familiar algebraic properties, it follows that the
scalar product of vectors has commutative and distributive
properties:
ab = b- a,
a(b + c) = ab + ac = (b + c)a
Determine the angle included between vectors
a = -i + 3j+k; b = 5i-4j-5k.
Let the sought angle be 0; then
0<6<%\ O°<0<18O°
The scalar product is
ab = -15-34-51 = -22

616
Chapter 16 VECTOR ANALYSIS
The magnitudes | a | and | b | are
|a| = V(-D2 + 32 + 12 = Vll,
|b| = V(5)2 + (-4)2 + (-5)2 = ^6.
cos 6 =
a • b
22
22
lal lbl V11 • 66 11^6 V6 '
0 = 2.526... rad ; 6 = 144.735 ... ° .
Return to the example on pp. 611 -12.
Find the angle 6 between cables AB and AC in the suspension
point, and the angle <f> between cable AD and the vertical
through point A.
x

Section 16.3 Multiplication of Vectors
617
We have
cos 0
rAB ' rAC
(rAB ) ^AC)
where we insert
rAB = -0.12 i - 0.12 j - 1.00 k rAB = V 1.0288
^AC = -0.12 i + 0.32 j - 1.00 k rAC = V 1.1168
Hence,
cos 0 =
(-0.12)(-0.12) + (-0.12)(0.32) + (-1.00)(-1.00)
(V 1.0288) (V 1.1168)
6 = 24.42°.
The vertical through A is the z-axis. We have
tad = 0.32 i - 0.12 j - 1.00 k rAD = V 1.1168 .
Hence,
COS 0 =
(0.32)0 + (-0.12)0 + (-1.00)(-1.00)
1.00 - V 1.1168
<p = 18.87°,
The Vector Product
The vector product, a xb, of three-dimensional vectors a and b
is a vector orthogonal to the plane that contains vectors a and b.
The vector product is also known as the outer product, or the
cross product, and reads "a cross b"„
We define a x b as the vector that fulfills the following criteria:
o a x b is perpendicular to the common plane of vectors a and
b, its direction being determined by the right-hand rule.
o The magnitude of a x b is | a | | b | sin 6, where 6 is the
included angle between the vectors a and b:
o-

618 Chapter 16 VECTOR ANALYSIS
Components of a Vector Product
In a three-dimensional orthogonal coordinate system the unit
vectors i, j, k are perpendicular to each other.
The magnitude of the vector product i x j is
|i| |j| sin 90° = 111 = 1
and its direction in a right-handed coordinate system is the
direction of the positive z-axis.
z
t
i k = ixj
l
x
The complete list of unit vector products is:
ixj = k jxk = i kxi=j
jxi = -k kxj = -i ixk = -j
ixi = jxj = kxk = 0
With
a = *ii+;yij+zik and b = ^2i + y2J+^k
the vector product expands as
axb = *iix*2i + *iixy2J+*iix*2 k+yij *x2i+yij xy2j
+ 3'lJX'Z2k + Zikx ^i+^ikx^j +zikxz2k,
which may be rearranged:
axb = (y\z2-y2Z\)i- (*1Z2-^2^i)J + (^i3,2-*2>i)k.

Section 16.3 Multiplication of Vectors
619
Properties of Vector Products
Since the vector product a x b contains the sine of the included
angle, it will be zero when a and b have the same direction and
will have its maximum value | a | | b | when the vectors are
perpendicular to each other.
A scalar may be placed anywhere in the vector product:
(Aa)xb = ax (/lb) = A (axb)
A vector product is distributive:
ax(b + c) = axb + axc
The right-hand rule decrees that the vector products cx = a x b
and c2 = b x a have opposite directions; that is, the vector product
Anti-commutative is not commutative, but anti-commutative,
axb = -(bxa).
Given: the vectors a = - i - j + 3 k and b = 5i + 5j-4k
Find the vector products axb and bxa, and their magnitudes.
axb = (-i-j + 3k)x(5i + 5j-4k)
lxi -5ixj + 4ixk-5jxi-5jxj + 4jxk
+ 15kxi+15kxj-12kxk
= -5x0-5k + 4(-j)-5(-k)-5x0 + 4i + 15j + 15(-i)-12x0
= -(lli-llj)
Solved with determinant:
p. 651

620
Chapter 16 VECTOR ANALYSIS
Since a x b = - (b x a), we have
bxa = Hi —11 j .
The results are illustrated in the figure below:
z
A
(-1,-1,3
The magnitudes of a x b and bxa are
|axb| = |bxa| = Vll2+ II2+02 = 11a/2
Solved with determinant:
p. 652
Given: a = -2i-2j+k; b = i + 2j+k
Find the vector product a x b.
axb= -4i+3j-2k
r I I If I ^-
The vector (- 4 i + 3 j - 2 k) is perpendicular to the plane
containing vectors a and b.

Section 16.3 Multiplication of Vectors
621
An architect decides to cut a corner of the top floor of his design
as shown in the figure below. The sloping part of the roof is to
be covered with sheet glass. For heat transfer calculations,
determine the area of the triangular cut OAB.
x
z
a
o
B (-20, 20, 0) m
—^y
■■■■I i(
^(0, 20,-10) m
Geometrically, the magnitude of the vector product a x b is the
area of the parallelogram determined by a and b.
The magnitude of the vector product of vectors a = OA and
b = OB is the area of a parallelogram with sides OA and OB,
b B
a = 20j - 10k
b = -20i + 20j
Solved with determinant:
p. 652
Since AB is the diagonal of the parallelogram, the sought roof
area OAB is half the magnitude of the vector product.
axb = 200i + 200j + 400k
|axb| = ^2002 + 2002 + 4002 = 200^6
AreaoAB = 2 laxbl = 10° ^6 * 245 m2.
There are other ways of calculating the sought surface area:
by using the Pythagorean theorem, by trigonometric methods,
and by Heron's formula.

622
Chapter 16 VECTOR ANALYSIS
Triple Products
Besides the scalar product and the vector product of two vectors,
there are several kinds of "mixed" products of the types
a • (b x c) and a x (b x c).
Solved with determinant:
p. 652
1
j
Scalar Triple Product
The combination a • (b x c) is defined as the scalar dot product
of a vector a and the cross product b x c of two vectors b and c.
If
a = xii+yij+z]k
b = x2i+y2J +z2k
c = ^3i+y3j+2;3k
then
bxc =^22:3-^3^2)1- (#2^3 -*3*2)J + (^2^3-^33^) k
and the scalar triple product
a-(bxc) = xi(y2z3-y3z2)-yi(x2Z3-X3Z2)+zi(x2y3-X3y2) •
The product of the three vectors a, b, c is positive if they form a
right-handed system; otherwise it is negative. Cyclic
permutation of the vectors a, b, c does not change the sign of the
product, whereas reversal of any two vectors reverses the sign
of the product:
a • (b x c) = b • (c x a) = c • (a x b)
= - a • (c x b) = - b • (a x c) = - c • (b x a)
Scalar and vector products of more than two vectors are not
defined in a general manner; only two vectors may be
multiplied with each other at a time, vector products taking
precedence over scalar products; these must be calculated first
as they are defined only between vectors.
Geometrically, the absolute value of a scalar triple product is
the volume of the parallelepiped with the edges a, b, and c:
bxc
x
The base of the parallelepiped is a parallelogram with sides
I b I, I c I, and the included angle 0. The area of the base is
bxcI = IbI I c I sin 0.

Section 16.3 Multiplication of Vectors
623
Since b x c is a vector product, it is perpendicular to the plane of
vectors b and c. The projection of a onto the b x c plane is the
height | a | cos 0 of the parallelepiped, whose volume is
| bxc| |a| cos0 = a-(bxc), ifO<0 <90°.
Find the volume of a parallelepiped defined by vectors
a = i + 2j-k
b = 2j+k
c = i-4j.
+ y
X
Solved with determinant:
pp. 652 - 53
a • (b x c) = 8 units of volume.
Vector Triple Product
The expression a x (b x c) is known as a vector triple product,
because the result represents a vector. That the result must be a
vector is evident from the fact that vector a is "crossed into" the
vector formed by b x c.
The position of the parentheses is crucial,
j x (j x k) = j x i = - k;
(jxj)xk = Oxk =0.
Since a vector product is perpendicular to the crossed vectors,
b x c is perpendicular to the plane of b and c; and a x (b x c) lies
in the be plane:
z
- y
a x (b x c)
x
Solved with determinant:
p. 652
If a = i + 2j-k; b = 2j+k; c = i-4j , findax(bxc)
bxc= 4i+j-2k;
ax(bxc) = -3i-2j-7k

624 Chapter 16 VECTOR ANALYSIS
§^rammmmmr!OTm^^
Qodcreated everything by weight, measure, and number.
The Apocryphal Book of Wisdom
From Sanctorius Sanctorius's Medicina Statica (first edition, 1614;
above illustration from an English edition printed in 1728).
Sanctorius spent long periods of time on a scale, making
careful registrations of his own weight and weighing all
intake and excretions. Through his observations, Sanctorius
made essential conclusions regarding evaporation from the
human body. Sanctorius's experimental research was greatly
influenced by the methods of his contemporary Galileo, who,
like Sanctorius, was at the University of Padua.
SANCTORIUS Sanctorius
(1561 - 1636) Italian physician
GALILEI Galileo
(1564 -1642)

625
Chapter
17
FRACTALS
Page
17.0 What Are Fractals? 626
17.1 The Snowflake Curve 627
17.2 Anti-Snowflake and Anti-Square Curves 629
17.3 The Cantor Set 630
17.4 Sierpinski Triangle, Carpet, and Sponge 631
17.5 The Mandelbrot Set 633
17.6 The Dimension Concept 635

626
Chapter 17 FRACTALS
17.0 What Are Fractals?
MANDELBROT Benoit B.
(b. 1924)
Latin, fractus, perfect
participle of frangere, "to break'
"How Long Is the Coast Line of Britain?" is a landmark
discourse in Science, presented in 1967 by the Polish-born French
mathematician Benoit Mandelbrot. At first thought, the
answer would seem to require measurements of photographs
taken from an orbiting satellite or, for greater accuracy, from
an aircraft flying along the coast or, for complete accuracy, by
walking along the coastline with a measuring tape.
However, the answers obtained will only be successively
closer approximations of the length of Britain's coastline. As
Mandelbrot pointed out, the length obtained depends on the
resolution of measurement, that is, the size of the smallest
bend seen on the photograph or measured on site.
Consequently, a coastline does not have a determinable length.
Analogously, nor does a river have a determinable length.
If we analyze the curve of a coastline or the course of a river,
we will find that, when magnified, parts of the curves are
identical with the whole, or nearly so; with a certain scale of
magnification, the pattern will repeat itself.
Similarly, the more a picture of a cloud is magnified, the more
we will be aware of a seemingly endless piling up of forever
smaller structures that repeat the general shape of the whole
cloud.
The universe is replete with shapes that repeat themselves on
different scales within the same object. In Mandelbrot
terminology, such objects are said to be self-similar.
In the idealized world of mathematics, there are several well-
defined figures that are self-similar and an infinite number
of such figures may be generated through iteration of
functions. These figures have quite unexpected properties, such
as a boundless perimeter enclosing a finite area, or a
boundless surface area containing a zero volume; the explanation
lies in the fact that these figures do not belong within the
human experience of a three-dimensional universe. The
word fractal - coined by Mandelbrot - was intended to
describe a dimension that could not be expressed as an integer;
today, "fractal" is generally understood to mean a set that is
self-similar under magnification.
Unlike mathematical fractals, no object in nature can be
magnified an infinite number of times and still present the
same shape of every detail in successive magnifications - one
reason being the finite size of molecules and atoms. Yet
fractal models may provide useful approximations of reality
over a finite range of scales.
Mandelbrot and others have applied fractals as explanatory
models of natural phenomena involving irregularities on
different size scales. This technique is used in graphical
analysis in such diverse fields as fluid mechanics,
economics, and lingnistics and in the study of crystal formation,
vascular networks in biological tissue, and population growth.

627
17.1 The Snowflake Curve
von KOCH Nils Fabian Helge
(1870-1924)
A "snowflake" curve is generated on the perimeter - a broken
line curve - of an equilateral triangle by the successive stages
of removing the middle third of a line segment and replacing
it by the other two sides.
This curve is also known as von Koch's curve, or von Koch's
island, named after the Swedish mathematician Helge von
Koch, who described it in 1904. An aim of von Koch's article
was to show that a simple geometric construction can lead to a
continuous curve that in every point is void of a tangent and
has no well-defined length.
The snowflake is a fractal; the self-similarity relates
successive stages: any portion of the curve at stage n will,
magnified by a factor of 3, look like a portion of the curve
at (n - 1).
The Perimeter
The perimeter of a snowflake curve increases indefinitely.
A2
Start with an equilateral triangle of side length a and area Aq.
The perimeter of the triangle is 3 a.
On the middle third of each side, construct a new equilateral
triangle of side length a/3 and area A\. The length of each
4
side of triangle Aq increases by a factor of 4/3, tora.
The fractal is the limit of successive stages; the perimeter of
the snowflake increases indefinitely towards infinity:
3a|-J
n
tends to oo ssn tends to <*>.
Thus, passing to the limit, the fractal has infinite perimeter.
A computer is helpful in the graphic display of fractals, but it
can only make a finite number of iterations and will never
show the completed fractal. The graph to the left depicts the 4th
iteration in the generation of a snowflake curve.
The snowflake curve is continuous, but a unique tangent
cannot be drawn anywhere on its periphery. It is continuous
everywhere but not smooth anywhere.

628
Chapter 17 FRACTALS
The Area
The area enclosed by the perimeter of the snowflake curve is
contained within the circumscribed circle of Aq, and thus the
snowflake curve encloses a finite area. The area approaches
an upper limit which is 8/5 the area of the original triangle:
A2
The sides of the original triangle Aq provide bases for three
new triangles Ai, thereby forming a six-pointed star whose
twelve sides will serve as bases for the next generation of
triangles A2, etc.
Each time a new triangle is added, it will provide four bases
for the following generation of triangles, and we have:
M = q Aq
A2 = (})2Ao
As = [q) Ao
XAi = 3-(1). A
o
Za2 = 3-4.fi) -Ao
1\
ZAs = 3-42-fi
J
Ao
A„ = [ gj Aq
XA„ = S-^-LfiV-A
m
0
and
oo
A=Ao + ^3.4»-i.(i)".Ao=A0 l4X(CT
n = 1
oo
n = 1
= A0
oo
1 +
! !•!(!)"
3"
+ 5_
= Ao
= 1^
[1 + 1 X
3' 1-4/9
3
n = 0
Ao
Thus, the snowflake curve has a perimeter of infinite length
enclosing a finite area. This curve and similar geometric
constructions were long looked upon mainly as mathematical
curiosities, but about 70 years after von Koch's description of
the snowflake curve, Mandelbrot used that curve and other
simple fractal constructions for his systematic description of
natural phenomena. What had been viewed as mathematical
oddities now became a means to describe reality!

629
17.2 Anti-Snowflake and Anti-Square Curves
The Anti-Snowflake Curve
If, unlike the von Koch curve, the triangles are pointed
inward, an anti-snowflake curve is generated. The
illustration below shows the first four iterations in the course of
generating an anti-snowflake curve:
Like the snowflake curve, the perimeter of the anti-snowflake
curve increases indefinitely, whereas its total enclosed area
approaches a finite limit that is 2/5 of the area of the original
triangle.
The proof is similar to that of the snowflake curve but, instead
of adding, we now subtract the triangles at each stage.
The Anti-Square Curve
An "anti-square curve" may be generated in a manner
similar to that of the anti-snowflake curve; the first four
iterations show the following pattern:
p

630
Chapter 17 FRACTALS
17.3 The Cantor Set
CANTOR Georg: p. 257
A Cantor set is generated by removing, in an iterative fashion,
a midportion of a true line segment.
The figure below illustrates a Cantor set generated by
removing the middle third of the original line segment [0, a] and
of every successive set.
0
a
1/3 a
1/9a 2/9a
2/3 a
7/9 a 8/9 a
When removing the midportions, we leave the endpoints
behind. In a way, a Cantor set may be regarded as the simplest
of all fractals; yet it offers several features to ponder.
In the Cantor set, we remove, from a segment of length a,
segments whose total length is a; yet we leave behind as many
points as we started with.
The whole is made up of two copies of itself at one-third scale.
A Cantor set contains no whole line segments, for at some
stage its midportion would have been removed, and in the end
we have totally disconnected points - a "dust".
Once thought to be mathematical curiosities, Cantor sets were
first applied by Mandelbrot as mathematical models for
solving problems of noise in electronic communication.
Using a solid instead of the one-dimensional line segment as
the initial stage, Mandelbrot has extended the concept of the
Cantor set to serve as a model of the distribution of matter in
the universe and as a model of the distribution of water
droplets in a cloud.
^tttor
Springing fiour white, rolling pie crust,
He thought about Mandelbrot dust.
fractals all-round the ledge
Qave the crust a fine edge.
Apple pie done just right was a must.

631
17.4 Sierpinski Triangle, Carpet, and Sponge
SIERPINSKI Vaclav The Sierpinski triangle, carpet, and sponge are named after
(1882 -1969) the Polish mathematician Vaclav Sierpinski.
The Sierpinski triangle, or gasket, starts out as an equilateral
triangle where the infinite succession of removals of
equilateral triangles takes place inside the triangle, the first
portion to be removed being a triangle with corners at the
midpoints of the sides of the original triangle. The three
equilateral triangular areas that remain within the original
triangle are each broken up into four equilateral triangles, of
which the central one is removed. This process may be
iterated ad infinitum and a fractal form is generated; the first
six iterations are shown here:
The Sierpinski triangle can be regarded as three copies of
itself, each at one-half scale.
The final Sierpinski triangle has an unbounded perimeter
and a zero area.
The Sierpinski carpet starts out as a square which is divided
into nine smaller squares, of which the central square is
removed. Each of the remaining squares is divided into nine
smaller squares from which again the central square is
removed. The process is iterated ad infinitum and a fractal
form is generated.
The Sierpinski carpet can be regarded as the union of eight
copies of itself, each at one-third scale.
Like the Sierpinski triangle, the Sierpinski carpet has an
unbounded perimeter and zero area.

632
Chapter 17 FRACTALS
MENGER Karl
(Austrian-born American
mathematician and philosopher;
b. 1902)
The Sierpinski sponge, or Menger sponge, is generated from a
cube in a manner analogous to the generation of the Sierpinski
carpet from a square. The cube is divided into 27 smaller
cubes, and the central cube and those at the center of each face
of the original cube are removed. Each of the 20 remaining
cubes is divided into 27 yet smaller cubes; the central cube and
the cubes facing the center of each face of the preceding cube
are removed.
When the process is iterated ad infinitum, a fractal form is
generated. Each external face of the Sierpinski sponge is a
Sierpinski carpet.
*>^.
1
The Sierpinski sponge is made up of 20 copies of itself, each at
one-third scale.
The Sierpinski sponge has zero volume, "enclosed" by an
unbounded surface area.

633
17.5 The Mandelbrot Set
JULIA Gaston Maurice
(1892 -1978)
French mathematician
Julia set of z2 + i.
The fractal behavior in the complex number plane is
demonstrated by iterating a nonlinear function whose variables
include its own result. If a set of an infinite sequence
f(z), flf(z)]> f if\f(z)]}• • •, where z is a complex number, is
plotted on a graph, the sequence of iterates may
1. be unbounded; or
2. jump around within a bounded region.
If (2) holds, we say that z lies in the "filled-in Julia set for f .
In the margin, the Julia set when f(z) = z2 + i is illustrated with
computer graphics.
The Mandelbrot set is related to the Julia set, but for it the
defining variable is the c in f(z) = z2 + c, where z and c are
complex numbers. Starting with z = 0 + Oi, we look for the
complex numbers c, such that 0,/(0), f\f(0)]9 ... remain bounded.
If we let z = 0 + Oi in f(z) = z2 + c, then
f(z) = A0 + 0i) = (0 + 0i)2 + c = c
flf(z)] = /(c) = c2 + c
f{flf(z)]} = f(c2 + c) = (c2 + c)2 + c
and the process may be iterated ad infinitum.
If c = 1 + i, the above functions yield
f(z) = /W + Oi) = 1 + i
f\f(z)] =/(1 + 1) = (l + i)2 + l + i = l + 3i
f{f\f(z)]} =/U + 3i) = [(1 + 1)2 + 1+ i]2+l + i
= - 7 + 7i and the iterates tend to «>.
We are now in a position to define the Mandelbrot set as the set
of all complex numbers c for which the iterated f(z) = z2 + c
remains bounded. The initial value of z is 0 + Oi and each
subsequent value of z is used to find the next one. Using
computer graphics, the Mandelbrot set is:
1.25-1
-2
-1.5
-1 -0.5
Real Axis
0
0.5

634
Chapter 17 FRACTALS
By zooming in on any of the outgrowths of the Mandelbrot set,
its self-similar fractal behavior is evident. The picture below
is a magnification of the □ area in the picture on the previous
page.
0.65-1
0.64-
0.09 0.10 0.11
0.12 0.13
Real Axis
0.14 0.15 0.16
DOUADY Adrien
(b. 1935)
HUBBARD John
(b. 1945)
Unlike the snowflake curve and the other fractals discussed
above, that of the Mandelbrot set is nonlinear; the difference
is that every self-similarity of a magnified Mandelbrot
offshoot is more obviously elaborated than its parent; that is,
Mandelbrot sets and Julia sets are self-similar in a common-
language way, but not in the strict scale-factor way of the Koch
snowflake.
The geometric link between the Mandelbrot and Julia sets is
that the Julia set for f(z) = z2 + c is disconnected when c lies
outside the Mandelbrot set. The Mandelbrot set itself is a
connected mass, as was shown by Adrien Douady and John
Hubbard. Thus, the elaborate tendrils of the Mandelbrot set
connect what appear here as islands.

635
17.6 The Dimension Concept
In Euclidean geometry a point has dimension zero; a shape
with length alone has dimension one; an area has dimension
two; a volume has dimension three.
Peano Curve
PEANO Guiseppe
(1858 -1932)
HILBERT David
(1862 -1943)
The Italian mathematician and logician Guiseppe Peano
described in 1890 a curve that could pass through every point of
a square; the construction of the Peano curve was simplified
by the German mathematician David Hilbert.
Divide a unit square into four sub-squares; join the centers of
the sub-squares by a broken line. Then divide every sub-
square of a unit square into four "sub-sub"-squares; join the
centers of all the sub-sub-squares by a broken line:
Proceed to the next stage by dividing a unit square into 4x4
squares and, again, join every center by a broken line; then
fashion 8x8 little squares and continue the process of drawing
a broken line:
fl
Fi
FW
€E
fcf*:!
FW
tm
m
EzftzJ
ilH
FW
"%1
Vz
Rft
fcRi
FW
Continuing the division of the square ad infinitum and every
time joining the centers of all the sub-sub-...sub-squares by a
continuous open curve, we approach a stage where the broken
line becomes a curve and passes through every point of the
plane.
This curve is not a fractal but does indeed present interesting
dimensional aspects. Tending to the limit of passing through
every point in a plane, the Peano curve may be argued to be of
two dimensions.

637
Chapter
18
MATRICES AND DETERMINANTS
Page
18.0 Scope and History 638
18.1 Matrices - Presentation 640
18.2 Matrices - Rules of Operation 642
18.3 Determinants 646
18.4 Special Matrices 654
18.5 Cofactors and the Inverse of a Matrix 659
18.6 Solving Systems of Linear Equations 661

638 Chapter 18 MATRICES AND DETERMINANTS
Once a mathematician named 2%^
Qardenedhis Cot, just for ftcfe.
And, as everything grows
In columns and rows,
He cheerfully weeded matrix^
18.0 Scope and History
Scope
A matrix is a rectangular arrangement of entries, displayed
in rows and columns. The entries may be any kind of
numbers, polynomials, or other expressions.
A square matrix has a determinant which has a value
determined by a rule of combination for the entries.
The value of the determinant may be used - either alone or in
combination with matrix operations - for solving systems of
linear equations, or for solving problems of figures displayed
in a coordinate system. Matrices are mainly used to solve
large systems of linear equations - discussed in this text -
and for transformation of objects in a coordinate system.
History
The ancient Chinese had a method of solving systems of
linear equations by representing the coefficients of the
unknown quantities by placing bamboo rods on a calculating
board in the same pattern as the entries of today's matrix.
This led to an early development of the principles of adding
and subtracting these rows and columns of the array.

Section 18.0 Scope and History
639
SEKI KOWA
(1642-1708)
von LEIBNIZ Gottfried Wilhelm
(1646-1716)
CRAMER Gabriel
(1704-1752)
VANDERMONDE
Alexandre-Theophile
(1735 -1796)
LAPLACE Pierre Simon de
(1749-1827)
LAGRANGE Joseph Louis
(1736-1813)
GAUSS Carl Friedrich
(1777-1855)
CAUCHY Augustin Louis
(1789-1857)
JACOBI Carl Gustav Jakob
(1804-1857)
HAMILTON William Rowan
(1805-1865)
Latin, matrix, "womb", "mould"
SYLVESTER James Joseph
(1814-1897)
CAYLEY Arthur
(1821-1895)
When Chinese science was introduced into Japan, the idea of
determinants soon gained acceptance. Seki Kowa, greatest of
17th-century Japanese mathematicians, in 1683 wrote Kai
Fukudai no Ho in which he discussed determinants and
methods of their expansion.
In 1693, the German Leibniz gave a formal description of
determinants as applied to the solution of systems of three
linear equations in three unknowns. In 1750, the Swiss
mathematician Gabriel Cramer established the general rule
for solving systems of n linear equations in n unknowns.
Alexandre-Theophile Vandermonde, in 1771, was the first to
recognize determinants as independent functions, apart from
their use for the solution of systems of linear equations. He
also described a number of properties of these functions, and
developed a form of notation more complete and appropriate
than that suggested by Leibniz. Vandermonde may be
regarded as the formal founder of determinant theory.
Laplace in 1772 described a general method of expanding
determinants in terms of their minors. Lagrange, in 1773,
handled second- and third-order determinants, and
introduced them for purposes besides the solution of equations.
Gauss, in 1801, used determinants in his theory of numbers
and suggested the idea of reciprocal determinants; he also
came very near the multiplication theorem.
The French mathematician Cauchy, in 1812, summarized
what was known of determinants at the time, and also
introduced the word determinant to mathematics. He started the
theory of determinants as a distinct, independent branch of
modern mathematics.
The next important contributor to the theory of determinants
was Jacobi, through whom the term determinant received its
final acceptance.
Thus, determinants were actually well established long before
the concept of matrices emerged. Although it had been implicit
in earlier writings, particularly by the Irishman William
Rowan Hamilton in 1839 and 1841, the term matrix was not
used until 1850 by the English mathematician James Joseph
Sylvester, who gave it its present meaning of a rectangular
array of numbers from which determinants may be formed.
Credit for understanding and identifying the full
significance of the algebraic properties of matrices is generally
given to the English mathematician Arthur Cayley who
presented fundamental work on matrices in 1855.
Because of many years of close collaboration, Cayley and
Sylvester are usually considered joint founders of matrix
theory, a recognition that should properly be shared by
Hamilton.
Matrices, and to a lesser degree determinants, find
application in physics and kindred sciences, in engineering,
economics, social sciences, and other fields where vast
volumes of data are handled.

640
Chapter 18 MATRICES AND DETERMINANTS
18.1 Matrices - Presentation
Entries, Elements
Rows Columns
A matrix is an array of real and/or complex numbers,
quaternions or any kind of numbers, known as entries, or elements,
disposed in a rectangular pattern of rows and columns.
The purpose of matrices is to provide a kind of mathematical
shorthand to facilitate the study of problems represented by the
entries: the matrices may represent transformations (linear
transformations) of coordinate spaces, or systems of
simultaneous linear equations.
Matrices are not numbers and have, in themselves, no specific
numerical significance but may be looked upon collectively
as a kind of higher-order number system carrying a built-in
code of interpretation embodied in the determinants of square
matrices.
In this text, we shall denote matrices by upright boldface
capital letters usually from the beginning of the Roman
alphabet, and enclose the matrices themselves by square
brackets; entries will be in lowercase italics.
Some texts use capital italics or German "Fraktur" characters
for matrices, enclosing them in parentheses or, less common,
double bars.
Row Index
Column Index
f
V
A
J
The entries of a matrix carry double subscripts; aij belongs in
the i-th row of the 7-th column, i being the row index andj the
column index. A matrix with m rows and n columns,
Dimension, Order
Square Matrix
Leading Entry, Leading Element
A = [atj]
m n
an
«21
«12
«22
«13
«23
«ln
«2ra
L am\ am2 ams
a
mn —'
is an mxn, or ra-by-ra, matrix; an reads a-one-one, not
a-eleven; a 12 is a-one-two; a2s is a-two-three, etc.
The expression m x n is the dimension, or order, of the matrix;
matrices with m = n are square matrices of the order m, or n,
denoted
Laij\ m \ L«i/J n •
The first non-zero entry in a row of a matrix is called the
leading entry, or leading element, of that row; in the matrix
below, 4 is the leading entry in the first row, 7 in the second
row:
"04-3
7 2
A =
[
n

Section 18.1 Matrices - Presentation
641
Principal Diagonal
Main Diagonal
Diagonal Entries
Vector
Algebraic Vector
Row Vector
Column Vector
Scalar
Lower Triangular Matrix
Upper Triangular Matrix
Trace
Scalar Matrix
Unit Matrix, Identity Matrix
E I
Null Matrix
The entries an, a22, «33, •••, a>nn that make up the principal
diagonal, or main diagonal - from upper left to lower right -
are diagonal entries.
If either m or n is unity, the matrix is a vector or, specifically,
an algebraic vector, to distinguish it from vectors represented
by directed line segments. The two forms of vectors are,
actually, different ways of expressing the same data.
A matrix with m = 1 consists of a single row vector, an
n-dimensional row vector; if n = 1, we have an m -dimensional
column vector:
an
X= [ an a12 a13 ... aln ] ;
Y =
«21
_ aml-
If m = n = 1, the matrix is reduced to a single entry and
becomes a scalar.
A matrix in which all entries above the main diagonal are
zero is a lower triangular matrix; if all entries below the
diagonal are zero, the matrix is an upper triangular matrix.
When all entries above and below the main diagonal are zero,
and all non-zero entries are diagonal entries, the matrix is a
diagonal matrix, and is usually denoted D.
A =
3
0
0
0
2
2
0
0
7
4
1
0
2 -
9
1
8 _
B =
" 7
8
5
_ 3
0
4
5
6
0
0
4
3
0
0
0
8
D =
3 0 0
0 12 0
L 0 0 4
Upper Triangular Matrix Lower Triangular Matrix Diagonal Matrix
The sum of the diagonal entries of a square matrix is the trace
of the matrix:
trA = 14 trB = 23 trD = 19
A diagonal matrix with an = a22 = «33 = ...= ann = k, where k is
a constant, is a scalar matrix. If k = 1, the matrix is a unit
matrix, E (for German "Einheit" = unity), or an identity
matrix, I.
0 0 0
1
I =
1
0
0 0
0 0 10
L 0 0 0 1 J
A matrix in which all entries are zero is a null matrix,
generally denoted 0 or by the Greek letter 6 (theta):
0 =
0
0
L 0
0
0
0
0
0
0
0
0
0 J
= 0
Null Vector
Analogously, a vector with only zero entries is a null vector.

Chapter 18 MATRICES AND DETERMINANTS
18.2 Matrices - Rules of Operation
Addition and Subtraction
Addition of matrices is defined only for matrices of equal
dimensions which have the same number of rows and
the same number of columns, that is, are conformable for
addition:
\_aij\mn + \yij\mn — laij + ^ijlmn
The actual operation of addition is performed by adding,
individually, corresponding entries of the matrices. Subtraction
is treated as negative addition.
A =
3
0
4
1
-2 7
6 4
0 1
3 0
B
2
4
1
2
4 3
6 0
7 4
3 1
A + B
5
4
5
3
2
12
7
6
10
4
5
1
A-B =
1
4
3
1
6
0
7
0
4
4
-3
- 1
The additive inverse of a matrix A is -A:
L o
-2 7
6 4
]
-A =
[
-3
0
2-7
6 -4
]
Multiplication
Multiplication by a Scalar
To multiply a matrix by a real or complex scalar k, every
entry a^ of the matrix is to be multiplied separately by k:
an
«21
ami
«12
a22
9
«m2
«13 •
a23
• •
» o
am3
■• aln
a2n
^m/i —
=
^aii
ka2i
•
- &ami
ka\2
ka22
«
«
kam2
kais
ka2s
© o
© •
kam%
tia\n
ka2n
©
ha
Matrix Products
The product of a row vector Aim and a column vector Bm i is
r &n
621 ' r i
= [ail&ll +^12^21 + ••• +«lm &mlj
[an ai2 ... aim]
L &ml ~

Section 18.2 Matrices - Rules of Operation
643
Two matrices A and B are conformable for multiplication, in
that order, if the number of columns in A is the same as the
number of rows in B.
The product of two matrices Amn and 1Bnpt
mn
an
«21
•
•
•
«ml
«12 •
a22 •
• •
• •
• •
am2 •
•• Glra
•• <*>2n
•
•
•
BMP =
&11 &i2 ... b\p
&21 &22 ••• b2p
- Kl bn2 .. b
np
ran* **np — ^mp
-aii&11+ai2&2l+-
a21&ll + «22 &21+-
is obtained by multiplying every row of A into every column
ofB:
+ alnbnl all &12 + «12 &22 + ■+alnbn2 ••• all by, + di2 &2p+- •+ aln b„p'
+ a2nbnl «21^>12+a22 &22 + -+a2nbn2 ••• a21&lp + a22 &2p + •■■ + a2n b„p
W
• • • •
Omlb\\+am2 &21 + >~+amnbnl «ml^l2 + am2^22 + ••• +amnbn2 ••• «ml^>lp + «m2^2p+•■•+ %n&
Matrix multiplication is distributive and associative,
A(B + C) = AB+AC; (A + B) C = AC+BC;
A(BC) = (AB)C,
but generally not commutative: the products AB and BA are,
as a rule, not equal.
A-[j-n=-[-i?]
Find AB and BA.
A--[j-n [-.'?]
~|_ 31 +
(- 2) (- 1)
4(-1)
1-0 +(-2) 1 If" 3-2]
30 + 41 J " L - 1 4 J
»*-[-in[ji]
r li + o.
" L (-1) i +1 ■
3 1 (- 2)
3 (- 1) (-2)
+ 04l[ 1 -2]
+ 1 - 4 J " L 2 6 J
The commutativity rule does apply, however, if one of the
factor matrices is the identity matrix:
AI = IA = A.
Any matrix multiplied by a null matrix will give the result
zero. There are also matrices that are not null matrices but
nevertheless will give zero products:
A =
0 10
10 1
0 1 0J
;B =
1-2 1
0 0 0
1 2 -1J
; AB = 0; B A * 0

644
Chapter 18 MATRICES AND DETERMINANTS
Premultiply
Postmultiply
Powers of Diagonal Matrix
In a product AB of two matrices A and B, B is said to be
premultiplied by A, and A to be postmultiplied by B.
The following rules apply for square matrices:
o Am AJ1 = Am + n
o (Am)n = Amn
A diagonal matrix raised to thep-th power remains diagonal.
Thus, if
^11 °12 °13 ••• °lrc ~>
°21 d22 023 ... 02n
D = O31 032 (I33 ... 03„
L 0
n\
0
n2
0
n3
dun
then
(D„)p =
Wn)p
021
031
012
w22)p
032
0l3
023
W33)P
L 0
n\
0
«2
0
n3
Oln "
o2«
o3„
D4of
is
D4 =
34
0
0
D =
0 0 -
24 0
0 74 .
" 3
0
. 0
=
0 0 -
2 0
0 7 _
" 81 0
0 16
_ 0 0
0 "
0
2401 _
The example below, simple to be sure, illustrates how one can
organize data into matrices.
A baker delivers one kind of bread. He charges $1.30 for a
single-loaf package and $2.20 for a double-loaf package.
The first week he delivers to a distributor 160 single-loaf
packages and 72 double-loaf packages; the second week 190
single-loaf packages and 80 double-loaf packages; the third
week 184 single-loaf packages and 100 double-loaf packages.
By the use of matrices, find - for each one of the three weeks -
the total number of loaves received by the distributor, and their
total value.
In the first row of a matrix A, indicate the number of loaves
in each package; in the second row, indicate the price per
package. Numbers involving double-loaf packages are italicized:
-[!.
30
2
2.20
]

Section 18.2 Matrices - Rules of Operation 645
In a matrix B, indicate in the first row the number of single-
loaf packages delivered weekly, and in the second row the
number of double-loaf packages delivered weekly,
[160 190 184 1
B " L 72 80 100 J *
[ 1 2 1 T 160 190 184 1
AB " L 1.30 2.20 A I 72 80 100 J
1x160 + 2x72 1x190 + 2x80 1x184 + 2x200
1.30 x 160 + 2,20 x 72 1.30 x 190 + 2,20 x 80 1.30 x 184 + 2,20 x 100
[ 304 350 384 1
" L 366.40 423.00 459.20 J
Week
Total Number of Loaves
Total Value $
1
304
366.40
2
350
423.00
3
384
459.20

646
Chapter 18 MATRICES AND DETERMINANTS
18.3 Determinants
Order
Main, or Principal, Diagonal
Secondary Diagonal
A determinant is a value representing sums and products of a
square matrix.
The determinant of a matrix A, det A, is denoted as arrays of
numbers or algebraic quantities, called entries or elements,
disposed in horizontal rows and vertical columns and
enclosed between single vertical bars:
det A =
an ai2
a in
• • » » »
an\ an2 ••• ann
Unlike matrices, which may be of any rectangular shape,
determinants are always square, having an equal number of
rows and columns called the order of the determinant; thus,
only square matrices have determinants.
The diagonal traversing the determinant from upper left
to lower right is the main diagonal, or principal diagonal;
the diagonal from lower left to upper right is the secondary
diagonal.
Every determinant represents a definite numerical value that
can be calculated according to specific rules which transform
the pattern of numbers into a single number that can be used in
calculations.
Determinants with real entries have real values, those with
complex entries have complex values.
The numerical value of an ra-th order determinant is the
algebraic sum of n! terms, each being the product of n different
entries taken one each from every row and column of the
determinant.
The value of a determinant of the first order is, of course, its
single entry; the value of a second-order determinant,
det A2 =
an a12
a2l a22
is the sum of its two permutations of two-entry products,
all x a22 (which reads a-one-one times a-two-two)
al2 x a2l (which reads a-one-two times a-two-one).
with their inversion correction factors (-1)* where k may be
measured by the number of inversions of the natural order of
the column index j,
a\\ c*22 digit order 1-2 no inversion (-1)° = 1
al2 a2l digit order 2-1 one inversion (-1)1 = -1,
and thus the determinant
detA2 =«11^22 ~a12a21-

Section 18.3 Determinants
647
«11 «22 «33
1-2-3
no. of inversions: none
correction factor: + 1
A third-order determinant,
detA3 =
«11 «12 «13
«21 «22 «23
«31 «32 «33
has 3! = 6 triple products of entries from every row and
column, of which no two triplets may be the same,
«11 «23 «32 «12 «21 «33 «12 «23 «31 «13 «21 «32 «13 «22 «31
1-3-2 2-1-3 2-3-1 3-1-2 3-2-1
two two three
one
one
+ 1
+ 1
-1,
and thus the determinant
detA3 = an (a22 «33 ~«23 «32)
- «12 («21 «33 ~ «23 «3l)
+ «13 («21 «32 ~ «22 «31 ) •
Example:
det A =
2
3
4
0 4
1 0
2 2
= 2(2-0)-0(6-0) + 4(6-4)
= 4-0 + 8 = 12
Sarrus's Rule
SARRUS J. P.
(1789 -1861)
A method of simplifying the development of a third-order
determinant - due to the Frenchman J. P. Sarrus - extends the
development of the determinant toward the right by repeating
the first and second columns:
«11 «12 «13
«21 «22 «23,
«31 «32 «33
positive products
«11 «12
«21 «22
«31 «32
«11 «12 «13
«21 «22 «23
«31 «32 «33
«11 «12
«21 «22
«31 «32
negative products
The first term of the development is the product an a22 «33 of
the main diagonal entries; the other two positive terms are the
products of the entries of the parallel diagonals in the
extension of the determinant. The three negative terms are
products of entries in the secondary diagonal and in the
diagonals parallel to it.
The use of Sarrus's rule is restricted to third-order
determinants.

648
Chapter 18 MATRICES AND DETERMINANTS
The Laplace Expansion
LAPLACE Pierre Simon de
(1749-1827)
Minor
First Minor
Principal Minor
Co factor
Inversion Correction Factor
Determinants of order three and higher are usually evaluated
by being developed into determinants of lower orders than that
of the parent determinant, by Laplace expansion into minors,
a method named after Pierre Simon de Laplace, French
mathematician, astronomer, and physicist.
A minor of a determinant is a sub-determinant obtained by
removing from the parent determinant an equal number of
rows and columns, so as to preserve the square shape.
Deleting the i-th row and the 7-th column from an n-th order
determinant leaves us with an (n - l)-th order first minor, or
principal minor, mij.
A cof actor cij of an entry aij is a first minor complete with its
appropriate sign or inversion correction factor (-1)1 +J9 which
is (+ 1) when the sum of the row index i and the column index j
is even, and (-1) when (i +j) is odd.
The determinant is the sum of all products of entries a^ and
their cofactors Cjy, complete with inversion correction factors.
Since this factor alternates regularly between (+ 1) and (-1), it
is sufficient to establish the sign for the first cofactor of an
expansion.
Determinants may be developed by the entries of any row or
column by the following procedure:
Select a row i or column,; of the determinant; the more zero
entries in it, the easier the calculation.
Multiply each entry of the chosen row, or column, by the
proper cofactor.
Continue this process until the expansion consists of only
second-order determinants.
0
0
0
Expand the determinant
det A =
by minors and determine its value.
0
1
0
3
4
0
6
5
-1
1
3
2
0
4
3
2
Since any row or column may be used for the expansion, we
select the first row because of its zero entries:
detA = 0(-l)<1 + 1) Imn |+■ 4 (-1)^1 + 2) \m12\ + (-l)(-l)(1 + 3) \m13 | + 0(-l)<1 +
= -4 \m12\ - |mi3|
114
4)
m14
I "*121 =
I mis I =
0 3 3
3 2 2
10 4
0 6 3
3 5 2
= 1(-1)(1 + 1)
3 3
2 2
= 1(-1)(1 + 1)
6 3
5 2
+ 1(-1)(1 + 2)
+ 0(-1)(1 + 2)
0 3
3 2
+ 4(-1)(1 + 3)
0 3
3 2
+ 4(-l)d + 3)
detA = -4(-27)-(-75) = 108 + 75 = 183
0 3
3 2
0 6
3 5
= -27
= -75

Section 18.3 Determinants
649
detA =
det AT =
detA =
detB =
Properties of Determinants
The value of a determinant remains unaffected:
- If all rows and all columns are transposed, without
changing entry order, so that rows become columns and columns
become rows:
«11 «12 «13
«21 «22 «23
«31 «32 «33
«11
«22 «23
«32 «33
«12
«21 «23
«31 «33
+ «13
«21 «22
«31 «32
= «11 («22 «33 ~ «23 «32) - «12 («21 «33 ~ «23 «3l) + «13 («21 «32 ~ «22 «3l)
«11 «21 «31
«12 «22 «32
«13 «23 «33
«11
«22 «23
«32 «33
"«12
«21 «23
«31 «33
+ «13
«21 «22
«31 «32
«11 («22 «33 ~ «32 «23) - «12 («21 «33
«11 («22 «33 ~ «23 «32) - «12 («21 «33
Example:
4
5
2
6
8
3
7
9
«31 «23) + «13 («21 «32 - «31 «22)
«23 «3l) + «13 («21 «32 ~ «22 «3l)
14 5
2 6 8
3 7 9
Corollary: Theorems stated for rows are also valid for
columns.
- If the entries of any one row, column, or a multiple of them
are added to, or subtracted from, another row, column, or
multiple, then the value of the determinant remains
unchanged.
Example:
12 3
4 6 7
5 8 9
12 3
4+2x1 6+2x2 7+2x3
5 8 9
The value of a determinant retains its numerical value but
reverses its sign if any two rows, or columns, are
interchanged, without changing entry order:
«11
«21
«31
«11
«31
«21
«12
«22
«32
«13
«23
«33
= «11 («i
«12
«32
«22
«13
«33
«23
«11
«22
«32
«23
«33
«12
«21 «23
«31 «33
+ «13
«21 «22
«31 «32
«12 («21 «33 ~ «23 «3l) + «13 («21 «32 ~ «22 «3l)
= «11
«32
«32
«33
«23
"«12
«31
«21
«33
«23
+ «13
«31
«21
«32
«22
= «11 («32 «23 ~ «33 «22) ~ «12 («31 «23 ~ «33 «2l) + «13 («31 «22 ~ «32 «2l)
~ [«11 («22 «33 ~ «23 «32) ~ «12 («21 «33 ~ «23 «3l) + «13 («21 «32 - «22 «3l) ]
Hence,
detB = -detA.

650
Chapter 18 MATRICES AND DETERMINANTS
Multiplying, or dividing, all entries of any one row, or
column, by the same scalar is equivalent to multiplying, or
dividing, the determinant by the scalar.
Corollary:
Example:
A factor that is common to all entries of a row, or
column, may be taken out and placed as a factor
in front of the determinant.
1
4
3x5
2
6
3x8
3
7
3x9
= 3
12 3
4 6 7
5 8 9
= 6
an 0 0
a21 a22 0
«31 a32 a33
If all entries above or below the principal diagonal are zero, the
value of the determinant is equal to the product of the entries of
the diagonal.
= an
«22 0
a32 a33
Example:
0
a2\ 0
a3l a33
1
4
5
5
1
4
0
-6
8
8
2
6
0
0
3
9
3
7
0
0
0
2
1
2
0
0
0
0
-4
6
0
0
0
0
0
8
+ 0
a21 a22
a31 a32
= «11^22^33
= 1(-6)3-2(-4)8 = 1152
Product of Determinants
det (A B) = (det A) (det B)
Example:
For
and their products
*» = [-5 "l7];B*:
U I ]
[
10 10
4 12
]
det A = 10; detB = 8; detAB = 80; detBA = 80.
Though multiplication of matrices is, generally,
non-commutative, the products of their determinants are equal.

Section 18.3 Determinants
651
Predicting Value Zero
The value of a determinant is zero if the entries of any one
row, or column, are
- all zero:
an a12 ai3
0 0 0
«31 «32 «33
= an
0
«32
0
«33
-«12
0
«31
0
«33
+ «13
0
«31
0
«32
= 0
equal, or proportional, to corresponding entries of
another row, or column:
«11 «12 «13
«31 «32 «33
«31 «32 «33
= «11
«32 «33
«32 «33
«12
«31
«31
«33
«33
+ «13
«31
«31
«32
«32
= 0;
«11 «12 «13
P «31 P «32 P «33
«31 «32 «33
px
«11 «12 «13
«31 «32 «33
«31 «32 «33
= px0 = 0
Geometric Significance
cf. unit vectors: p. 607
pp. 617 et seq.
cf. p. 620
The geometric meaning of the absolute value of a second-order
determinant,
detA =
«ll
«21
«12
«22
is that it gives the area of a parallelogram spanned by vectors
ai and a2 whose coordinates are the columns (or rows) of A; the
magnitudes of the vectors are taken relative to the unit vectors
of the system. Similarly, the absolute value of a third-order
determinant gives the volume of a parallelepiped.
The vector product of
a = xii+yij+zik and b = X2i+y23 +^2^
may be expressed as
axb =
i J k
*i yi zi
*2 yi *2
Examples:
The products of vectors
a = -i-j + 3k and b = 5i + 5j-4k
are
axb = (-i-j + 3k)x(5i + 5j-4k)
= -5ixi -5ixj+ 4ixk-5j xi-5j xj + 4j xk
+ 15kxi+15kxj-l2kxk
= -5x0-5k + 4(-j)-5(-k)-5x0 + 4i+15j + l5(-i)-12x0
= -(lli-llj); bxa=lli-llj.

652
Chapter 18 MATRICES AND DETERMINANTS
The products a x b and b x a may, more conveniently, be
obtained from the determinants:
axb =
iik
i i q .-13 .-13 ,-1-1
l~l 3A =1 5-4 "J 5-4 +k 5 5
5 5-4
= i(4-15) -j(4-15) +k(-5 + 5) = -(lli-llj)
and
bxa =
i J k
5 5-4
-1 -1 3
= l
5 -4
1 3
5
-1
•4
3
+ k
5 5
1 -1
cf. p. 620
= i(15-4) -j(15-4) +k(5-5) = lli-llj
Similarly, a x b of vectors
a = —2 i — 2 j + k and b = i + 2j +k
is
axb =
i J k
-2 -2 1
12 1
= i (-2 - 2) - j (-2 - 1) + k (-4 + 2)
= -4i + 3j-2k
p. 621
In the problem on p. 621, the vector product of
a = 20j-10k and b = -20i + 20j
may be conveniently determined if written
axb =
i J k
0 20 -10
-20 20 0
= {(0 i + 200 j + 0 k) - [0 j + (- 200 i) + (- 400 k)]}
= 200 i + 200j +400 k.
p. 622
For vectors
a = x\ i + y\ j + Z] k >
b = x2 i + 3>2 J + ^2 k ),
c = x3i+y3j+z3k ^
we have the scalar triple product
a-(bxc) = ^1(3/2^3-^3^2)-^1(^2^3-^3^2)+^1(^2^3-^3^2),
which we may write
a • (b x c) =
*i y\ *i
X2 y2 z2
*3 ys *3
p. 623
Examples:
The scalar triple product a • (b x c) of vectors
a = i + 2j-k; b = 2j+k; c = i
represents the volume of a parallelepiped
-4j

Section 18.3 Determinants
653
p. 623
cf. p. 552
a • (b x c)
1
0
1
2 -1
2 1
-4 0
8 units of volume.
For the vector triple product a x (b x c), we first calculate
bxc
i J k
0 2 1
1-4 0
= 4i+j-2k,
then:
a x (b x c)
i J
1 2
4 1
k
- 1
-2
3i-2j-7k
The area A of a triangle placed in an orthogonal coordinate
system may be conveniently expressed in notations of the
absolute value of a determinant, where the entries of the third
column are unity:
1
2
*i y\ i
*2 yi i
*3 yz i
(xl9y{i
Example:
(*3> 3¾)
x
(*2> 3¾)
If the vertices of a triangle are at (3, -2), (-4, 1), (8, 3) in an
orthogonal coordinate system, the area A of the triangle is
-i
3
-4
8
2 1
1 1
3 1
-(3-16-12)-(8 + 9+8)
= 25 square units.

654
Chapter 18 MATRICES AND DETERMINANTS
18.4 Special Matrices
The Transpose Matrix
Symmetric Matrix
Skew-Symmetric Matrix
A =
B =
[
[
If a matrix A is reflected in its main diagonal, so that all rows
become columns and all columns become rows without
changing their relative order or the order of entries in the rows
and columns, the result is a transpose matrix, AT:
A =
[
«11 «12 «13
«21 «22 «23
]
AT
«11 «21
«12 «22
L «13 «23
A matrix A is:
- symmetric if A = AT
- skew-symmetric if A = -AT
The matrix
4-12
-117
2 7-1
4-12
117
2 7-1
is symmetric, since
4-12
117
2 7 -1 J
-■ T
0-4 7
4 0 3
■7 -3 0
is skew-symmetric, since
0-4 7
4 0 3
-7-3 0 J
0-4 7
4 0 3
L-7 -3 0
-, T
The following calculation rules apply for transpose matrices:
o (AT)T = A;
o (k A)T = kAT, where k is a scalar quantity;
o (A+B)T=AT + BT, if A and B are conformable for
addition.
Example:
A + B =
[
3
0
2
4
5
4
-2 7
6 4
4 3]
6 -1 J
2 10
12 3
]
AT =
BT =
" 3
-2
_ 7
r2
4
L 3
0 1
6
4 J
4 "I
6
-1 J
]
(A+B)T
5
2
10
4 I
12
3 J
AT + BT =
r 5
2
L io
4
12
3

Section 18.4 Special Matrices
655
O
(AB)T = BTAT,
if A and B are conformable for
multiplication, in this order.
Example:
A =
B =
[
[
3
0
2
4
-2
24
2 1
6 J
4 3
6 -1
0
36
]
11
-6
r -2
0
L li
24 1
36
-6 J
AT =
BT =
BTAT =
r3
L -2
r 2
4
_ 3
" -2
0
_ 11
0
6
4 "
6
-1 .
24
36
-6
]
AB =
[
]
(AB)T =
Analogously, (A B C )T = CT BT AT if A, B, C are conformable
for multiplication, and (ABCD)T=DTCT BT AT if A, B, C, D
are conformable for multiplication, etc.
Inverse Matrices
For the inverse A-1 of a matrix A, we have
A A1 = A1 A = I.
A matrix has an inverse only if its determinant ^ 0.
Why? det (A • A"1) = det (A) • det (A"1) and det 1=1.
If det A = 0, we have
0 • det (A"1) * 1,
and A • A"1 = I does not exist for A.
Only square matrices have inverses.
The product of the square matrix An and its inverse A^_1,
A„ =
an ai2 .
«21 «22 •
• • •
• • •
- anl an2 •
■• aln
■• a2n
•
•• ann —
A -1 -
*11 *12
*21 *22
• •
• •
— xnl xn2
... X\n
• • • %2n
• •
• •
• •
• • • xnn
IS
auxu + ai2x2i+ ...+ alnxnl an x12 + a12x22 + • •• + ainxn2
a2ixn + a22x2i + ...+ a2nxni a2ix12+ a22x22+ ...+ a^ xn2
an xin + a12 x2n + ... + aln xnn'
a21 *ln + «22 x2n + ■ • • + «2n ^nn
anl xll + an2 x2l +---+annxnl anl xl2 + an2 *22 +--• + ann ^2 ••• °fol *ln + «n2 *2n + • ■ • + a>nn xnn

656
Chapter 18 MATRICES AND DETERMINANTS
Since
we have
A A -1 - T -
1 0
0 1
0 0
0 1
0
all *11 + a\2 *21 +• • • + a In xnl
all *12 + a12 *22 + • • • + CL\n Xn2
= 1
= 0
all xln + a12 x2n + • - + aln xnn =0
^21^11+^22^21 + ---+^2^^1 = 0
a21 *12 + a22 *22 + • • • + a2n xn2 = 1
a21 xln + a22 x2n + • • •+ a2n xnn = ®
an\ x\\ + an2 *21 +--+ CLnn xnl = 0
an\ x\2 + an2 ^22 + • • • + ann xn2 = 0
Singular Matrix
Invertible or Nonsingular Matrix
+ ann xnn
= 1
Gnl x\n + ^n2 x2n +
A square matrix for which there exists no inverse - that is,
a square matrix whose determinant is 0 - is known as a
singular matrix; a matrix for which there exists an inverse
(determinant * 0) is an invertible or nonsingular (irregular)
matrix.
Find the inverse matrix of A =
12 0
0 3 8
L 1 0 -5
|x(2)
(3)-(1)
MMMWMWMMMVMVMWMMWWWMMMWMMWWi
(3') + 2 x (21)
3 x (3M)
8
^MMMWWWWVMAMMWMWMMAMMWWWVWMWAWWMAMAMMAMh
(2') - 5 x (3'")
(1) - 2 x (2")
MWIAMNHMtmNWININVlW
(1)
(2)
(3)
(1)
(2')
(3)
(1)
(2')
(3')
(1)
(2')
■ (3M)
(1)
(2')
(3,M)
(1)
(2")
(3'M)
-(11)
(2")
(3,M) L
1
0
1
1
0
1
1
0
0
1
0
0
1
0
0
1
0
0
1
0
0
2 0
3 8
0 -5
2 0
1 8/3
0 -5
2 0
1 8/3
-2 -5
2 0
1 8/3
0 1/3
2 0
1 8/3
0 1
2
1
0
0
0
1
0 0
1
0
0
1
10 0
0 10
0 0 1
10 0
0 1/3 0
0 0 1
10 0
0 1/3 0
-1 0 1 J
10 0
0 1/3 0
-1 2/3 1 J
10 0
0 1/3 0
-3 2 3 J
10 0
8 -5 -8
-3 2 3 J
-15 10 16
8 -5 -8
-3 2 3

Section 18.4 Special Matrices
657
A"1 =
f -15 10 16
8 -5 -8
-3 2 3
If B = I o 4|» ^11^ the inverse matrix B_1.
WithB"1 =
[
xi
x3
x2
*4
I , we have, since BB-1 = I a 1
ri 2irx! x2 l r
L 3 4 J L JC.q Xa J " L
x3
xi + 2 x3 = 1
3 jci + 4 X3 = 0
xi + 2 X3
3 xi + 4 X3
*2 + 2 jc4 "I
3 JC2 + 4 X4 J
JC2 + 2x4 = 0
3 x2 + 4 X4 = 1
-[j?i
Coefficient matrix and
augmented matrix are further
described on p. 662.
As the coefficient parts of the augmented matrices
[3 4 0 ] ^ [ 3 4 l]
4 0 ] and [ 3 4
are identical, we may augment the parent matrix by the
identity matrix and reduce the elimination procedure to only
one matrix:
[1210] [1
L3 4 0 lJ^Lo
2
-2
1
-3
0] [10-2 11
1 J =* L 0 1 3/2 -1/2 J
^ ~ L 3/2 -1/2 J
The following calculation rules apply if A is invertible:
0 (A"1)"1 = A;
0 (A71)-1 = (A-1)71 , where n is a positive integer;
0 (AT)"1 = (A"1)1";
o (A B)_1 = B_1 A-1, if A and B are invertible and
conformable for multiplication;
0 det(Am)"1 = 1/detA,
Lm
Orthogonal Matrix
Orthogonal Matrices
A matrix A that is equal to the inverse of its transpose matrix,
(AT)~ !, is an orthogonal matrix.
Example:
A =
AAT =
1
V2
1
- V2
1
V2
1
- V2
1
V2
1
V2 _
J_
V2
J_
V2 J _
is orthogonal, and therefore
1
V2
1
V2
1
V2
1
V2 _
-Vol]
If two matrices A and B are orthogonal and conformable for
multiplication, their product AB is also an orthogonal matrix.

658
Chapter 18 MATRICES AND DETERMINANTS
Complex Matrices
Complex Conjugate Matrix
Adjoint Matrix
Hermitian Conjugate Matrix
Normal Matrix
Unitary Matrix
Hermitian Matrix
HERMITE Charles
(French mathematician;
1822-1901)
Projection Matrix
Skew-Hermitian Matrix
In a complex matrix, at least one of the entries is a complex
number. If the complex entries of a square matrix A,
2 3-i 4
A = 2 + i 7 -i
4 i 0
are replaced by their conjugates, the new matrix is a complex
conjugate matrix,
r 2 3 +i 4
A= 2-i 7 i
4 -i 0
the transpose of which is an adjoint matrix,
2-i 4
A* =
2
3 + i
4
7
l
-l
0
also called an Hermitian conjugate matrix.
If A A* = A* A, then A is known as a normal matrix.
A unitary matrix, U, has the property U U* = I, that is, U* is
U_1. The unitary matrix is normal.
A matrix that is its own Hermitian conjugate is an Hermitian
matrix.
is Hermitian, as A = A*.
2 3 + i 4
A = 3-i 7 i
4 -i 0
The Hermitian matrix is a normal matrix.
An Hermitian matrix P such that P2 = P is known as a
projection matrix.
A matrix such that A* = - A is a skew-Hermitian matrix.
The purpose of mentioning the various subgroups of complex
matrices is only for reader awareness; they will not be met
again in this text.
from his days at yak, long -past, the sf^eiv Hermitian haunted'Bill,

659
18.5 Cofactors and the Inverse of a Matrix
p. 648
A cofactor of an entry of a matrix is the same as the cofactor of
the same entry in the determinant of the matrix and, thus, is
defined only for square matrices.
Example:
Find the cofactor matrix of A =
Let
12 0
2 14
L 4 2 6
det (cofactor matrix of A) =
An A12 A13
A2i A22 A23
^31 ^32 ^-33
where
An = +
1 4
2 6
= (1x6-4x2) = -2
A12 = -
2 4
4 6
= -(2x6-4x4) = 4
Aqi =
L21
^13 = +
2 0
2 6
2 1
4 2
= (2x2-1x4) = 0
= -(2x6-0x2) = -12
A22 = +
1 0
4 6
= (1x6-0x4) = 6
Aqi = +
L31
A23 =
2 0
1 4
1 2
4 2
= -(1x2-2x4) = 6
= (2x4-0x1) = 8
A32 =
1 0
2 4
= -(1x4-0x2) = -4
^33 = +
1 2
2 1
Hence,
cofactor matrix of A =
= (lx
A =
1-2x2) = ■
" -2 4
-12 6
_ 8 -4
-3
0
6
-3
The transpose of the cofactor matrix may be used to find the
inverse of a matrix, if it exists. From cofactor expansion and
product properties of matrices, we have
det A 0 ... 0
0 det A ... 0
A (cofactor matrix of A)T =
0
0
detAJ
and by including the factor 1/det A ,
(1/det A) A (cofactor matrix of A)T = I

660
Chapter 18 MATRICES AND DETERMINANTS
By definition A • A"1 = I, so, if A"1 exists,
1
detA
A (cofactor matrix of A)T = A • A 1,
and we have a formula for A"1:
A"1 =
detA
(cofactor matrix of A)T
Example:
To find A"1 of A =
12 0
2 14
4 2 6
cofactor matrix of A =
(cofactor matrix of A)T =
, we have from the above that
-2 4 0-
-12 6 6
8 -4 -3 .
-2-12 8
4 6-4
0 6-3
(
let A =
Hence,
A-i-l"
A " 6
1
2
4
2 0
1 4
2 6
-2 -12 8
4 6-4
. 0 6 -3 .
= 6
=
>.
--1/3
2/3
_ 0
Checking:
AA"1 =
" 1 2
2 1
_ 4 2
0 -
4
6 _
--1/3
2/3
_ 0
-2
1
1
4/3 "
-2/3
-1/2 _
-2 4/3 "
1 -2/3
1 -1/2 _
=
" 1 0
0 1
_ 0 0
•
0
0
1
= I

661
18.6 Solving Systems of Linear Equations
We shall discuss three methods:
- Elementary row operations
- Use of the inverse of a matrix
- Use of determinants (Cramer's rule)
Elementary Row Operations
Elementary Operations
By Gaussian elimination, or reduction, a system of linear
equations is transformed into an equivalent system in
echelon form, which is easy to solve by successive back
substitution.
The transformation to echelon form is carried out by one or
more of the following elementary operations:
o
o
o
Interchange equations;
Multiply (or divide) an equation by a non-zero quantity;
Add or subtract any equation, or multiple of it, from
another equation of the system.
Consider the system of
equations:
1. Interchange Equations
(1) and (3):
2. Divide Equation (3)
by 2 to make the
coefficient of x unity:
3. Subtract Equation (2)
from (1):
4. Divide Equation (1)
by 3 to obtain the
echelon form:
(1)
(2)
(3)
(3)
(2)
(1)
(3')
(2)
(1)
(3')
(2)
(D
(3')
(2)
(1")
y + 4z = 6 >
y + z= 3 >
2x+4y+6z = 20 ^
2x + 4y + 6z = 20 "
y + z = 3
y + 4z = 6^
x+2y+3z = 10 "
y+z= 3 I
y + 4z = 6
x + 2y + 3z = 10 ^
y+z = 3
3z= 3
>
j
x + 2y + 3z = 10 ^
y+z = 3
z= 1
>
j

662
Chapter 18 MATRICES AND DETERMINANTS
JORDAN Camille
(1838-1922)
When the procedure is carried on until the system of equations
is completely solved, the systematic method is called Gauss-
Jordan elimination, or reduction.
Continuing from No. 4 above:
5. Multiply (2) by 2 and
subtract from (3'):
6. Subtract (1") from (3")
and (1") from (2):
(3")
(2)
(1")
(3'")
(2')
(1")
x + z =
y + z =
z =
X =
y =
z =
4 ^
3
1
>
j
3 ^
2
1
>
j
Coefficient Matrix
Augmented Matrix
Systems of linear equations can be advantageously adapted
for computer programming by being presented in the form of
matrices containing the coefficients of the variables and the
constant terms but leaving out the variables. This simplified
system is permitted since the Gaussian and Gauss-Jordan
elimination methods involve only arithmetic operations on
coefficients and constant terms.
Instead of using different letters (x, y, z) when handling
systems of equations involving several variables, it is more
convenient to use one letter with indices (jci, x2, ..., xn).
A system of m linear equations in n unknowns,
an xi + a12 x2 + ... + a\n xn = c\ >
a2i X\ + a22 #2 + ---+ a2n xn = c2
>
am\ x\ + am2 #2+---+ amn xn = C
m
J
can be expressed by the coefficient matrix
an al2
a21 a22
L am\ am2
and the augmented matrix
^2n
amn —
an
a>2\
«12
a22
ain
G2n
L flml am2
a
mn
C2
cm —
which also includes the constant terms.
Every row of the augmented matrix must contain coefficients
for all of the unknowns of the equation system; unknowns that
are missing in an equation are assigned the coefficient 0.

Section 18.6 Solving Systems of Linear Equations
663
Elementary Row Operations
Pivot Operation
Pivot Entry
Pivot Row
Target Row
o
o
The augmented matrix is simplified by a series of elementary
row operations - a variation of Gauss-Jordan reduction which
say that
rows may be interchanged;
the entries of any one row may be multiplied by a non-zero
constant;
o any multiple of a row may be added to another row.
Using these operations, every square matrix may be
transformed into a triangular matrix.
Each step in this reduction process consists in making a
selected matrix entry aij into a "1" and using it to produce zero
entries preceding it in the i-th row and above and below it in
they-th column.
This procedure is known as a pivot operation, or pivoting for
short; a/y is the pivot entry, or pivot, which must, of course, be
non-zero. The row with the pivot entry is called the pivot row;
a row to be transformed is a target row.
Example:
Pivoting about the second entry of the second row (pivot entry
in boldface) gives
- 1
0
_ 2
=>
8 3 2"
2 4-1
4 1 8 _
"18 3
0 12
_ 2 4 1
=>
2 "
-1/2
8 _
" 1
0/2
_ 2
8
2/2
4
3
4/2
1
2
-1/2
8
1_0 8-8-1 3-8-2 2-8 (-1/2)
0 1 2 -1/2
2-0 4-4-1 1-4-2 8-4 (-1/2)
" 1 0 -13 6
0 12 -1/2
_20-7 10
Linearly Independent
Linearly Dependent
Existence of Solutions
Linear independence of two equations, or rows of a matrix,
means that there exists no proportionality between the
equations or between the rows.
For the linearly independent binomials
x + 2y ; 2x+y ; x andy not both = 0
we have
-(x + 2y) + 2x+y * 0;
for the linearly dependent
x + 2y\ 2x + 4y,
-2(x + 2y) + 2x + 4y = 0.

664 Chapter 18 MATRICES AND DETERMINANTS
Rank The rank of a matrix is equal to the dimension of the largest
sub-matrix that can be obtained by deleting rows and columns
of the parent matrix and that has a non-zero determinant.
It is equal to the number of linearly independent rows (or
columns) of a matrix in echelon form that do not have
exclusively zero entries.
The ranks of the coefficient and augmented matrices may be
used to establish the consistency of the equation system and the
existence of solutions of the equation.
The equations of a system of linear equations are solvable
if the matrix formed from the coefficients of the variables,
p 662 ^e coefficient matrix, has the same rank as the augmented
matrix.
Finding the Solutions
A matrix reduction may typically be carried out according to
the procedure described below.
o Interchange rows, if necessary, so as to make the first row
begin with a non-zero entry which is made into a "1" by
division.
o Add a suitable multiple of the first row to the second row in
order to create a zero in the first position of the second row;
this is equivalent to removing one unknown from the
corresponding equation.
o Make the leading entry of the transformed second row
into a "1" and pivot the matrix about this entry.
o If the intended next pivot entry turns out to be a zero,
change this row for another from below it, in order to obtain
a non-zero pivot entry.
o Proceed in this manner down the matrix till the last row;
all new zero entries correspond to the successive
elimination of unknowns in conventional algebra.
o If it proves impossible to obtain a non-zero pivot entry, the
reduction procedure is finished, and the matrix is in
echelon form.
Solve
(1) y + 2z = -5"
(2) 5x+y-Sz = 15 >
(3) Sx+y-2z = 5

Section 18.6 Solving Systems of Linear Equations
665
Write the system
as an augmented
matrix.
Interchange rows
(1) and (2).
Pivot about the
first entry in row
(2).
Pivot the matrix
about the second
entry in
row (2):
Pivot the matrix
about the third
entry in
row (2).
Hence,
2)
1)
3)
2)
1)
3)
2)
1)
3)
2)
1)
3)
2)
1)
3)
x = 6; y = -9; z = 2
1)
2)
3)
2)
1)
3)
0
5
L 3
1
1
1
2 -5
-3 15
-2 5
5 1 -3 15
0 12-5
3 1 -2 5 J
2)
1)
3)
1
0
_ 3-3
1/5
1
1-3/5
1 1/5 -3/5
0 12
L 0 2/5 -1/5
-3/5
2
-2 + 9/5
3
-5
-4 J
3
-5
5-9
1 1/5 - 1/5 -3/5 -2/5 3 + 1
0 1 2-5
0 2/5 - 2/5 -1/5 - 4/5 -4 + 2
10-14
0 12-5
L 0 0 -1 -2 J
1 0-1 + 1 4 + 2
0 1 2-2-5-4
0 0 1 2
10 0 6
0 10-9
L 0 0 1 2 J
This solution follows the strict Gauss-Jordan elimination
method, suitable for computer programming. By employing
elementary row operations more freely, one might force a pivot
entry to be 1, but this can result in cumbersome fractions
elsewhere; the strict elimination method is always the most
dependable.
Use of the Inverse of a Matrix
When several systems of simultaneous linear equations have
the same coefficient matrix, the inverse of this matrix
suggests itself as an efficient means of solving the systems.
Finding the inverse of a matrix is a laborious process, so this
method has no advantage over other methods when solving an
isolated system of equations. If a computer or a scientific
calculator with matrix capacity is available, the inverse of a
matrix may be readily found and conveniently used for any
system of linear equations.

666
Chapter 18 MATRICES AND DETERMINANTS
A system of linear equations may be written in matrix form,
AX = C,
or, after premultiplication by A-1,
A"1 AX = A-iC
IX = A-!C
X = A-iC
where
- A is an invertible coefficient matrix;
- A"1 is its inverse matrix;
- X is the column vector of unknowns;
- C is the column vector of constant terms.
Matrices used here are smaller than generally encountered in
practical applications.
Solve the system
xi + 2 X2 = 4 >
2 xi + X2 + 4 *3 = 3 >
4 xi + 2 X2 + 6 x% = 2
A =
1
2
4
2
1
2
0 -
4
6 _
; x =
Xi
x2
-*3
c =
4
3
L 2
pp. 655 - 57
A"1 =
-1/3 -2 4/3
2/3 1 -2/3
0 1 -1/2
X = A-iC
Xi ~
x2
-*3 -
—
- -1/3 -
2/3
_ 0
--14/3 -
13/3
_ 2 _
-2 4/3 "
1 -2/3
1 -1/2 _
" 4 "
3
_ 2 _
2 1
*i = -4t; *2 = 4t; *3 = 2
The baker met earlier still charges $1.30 for a single-loaf
package and $2.20 for a double-loaf package. This time
information is available about the total number of loaves
delivered during each of three consecutive weeks and the
weekly values of the sales:
Week
4 5 6
Total Number of Loaves
360
416
420
Total Value $
432.00 492.80 502.00

Section 18.6 Solving Systems of Linear Equations 667
How many packages of each kind were sold per week?
Let jcn be the number of 1-loaf packages sold in the 4th week,
JC12 the number in the 5th week, and xi% the number in the 6th
week.
Analogously, let #21, #22» anc^ x23 be the numbers of the 2-loaf
packages sold in the 4th, 5th, and 6th weeks, respectively.
We have the equations
jqi + 2 X21 = 360
1.30jcii + 2.20^21 = 432.00
*12 + 2 X22 = 416
1.30jci2 + 2.20jc22 = 492.80
*13 + 2 x23 = 420
1.30jci3 + 2.20jc23 = 502.00
The coefficient matrix A,
A-l"1 2 1
A ~ L 1.30 2.20 J '
displays in the first row the number of loaves of bread and in
the second row the cost.
The constant terms matrix C,
[360 416 420 "I
C " L 432.00 492.80 502.00 J '
displays in the first row the total number of loaves for each
week and in the second row the total weekly sales.
Let a matrix X contain the sought numbers:
r xn xi2 X13 1
L Xo-l X99 *23 J '
A"1 =
1x2.20 -2x1.30
r 2.20 -2]
L -1.30 1 J
""0.4 L-1.30 1 J ~ L 3.25 -2.50 J
Since A X = C, which gives X = A"1 C, we have
Xn Xi2 X13
*21 *22 *23
"I J_ |~ 2.20 -2] [360 416 420 "I
J ~ " 0.4 L -1.30 1 J L 432.00 492.80 502.00 J
[ 180 176 200 "I
" L 90 120 110 J '
Week
4 5 6
Total Number of 1-Loaf Packages 180 176 200
Total Number of 2-Loaf Packages 90 120 110

668
Chapter 18 MATRICES AND DETERMINANTS
Cramer's Rule
CRAMER Gabriel
(1704-1752)
p. 660
p. 666
Cramer's rule - named after the Swiss mathematician and
physicist Gabriel Cramer - uses determinants to express the
solution of a system of linear equations.
For an invertible matrix A, we have
1
A"1 =
detA
(cofactor matrix of A)T
and, since X = A"1 C ,
1
X =
det A
(cofactor matrix of A)T C ,
that is,
xi
x2
• • •
_ Xm _
det Am
An Ml
^•12 ^-22
^ml
Am2
L A
Ira
■2m
mm -1
Cl
C2
• • •
_ Cm -
For a system with two linear equations in two unknowns, this
reduces to
ai 1^1+012^2 = cl
a2i*i+a22*2 = c2
The coefficient matrix and the transpose of the cofactor matrix
of A are
A =
an ai2
«21 «22
; (cofactor matrix of A)T =
«22 -«12
-«2l an
and thus,
[*1 I 1 a22 -«12 1 |"ci "I
X2 J det A L _a2l «ll J Lc2 J '
which gives
xi =
x2 =
(«22 cl ~Ql2c2)
detA
(a2ici -anc2)
detA
(aiic2-a2ici)
detA
whose numerators can be written as determinants
ci ai2
C2 «22
Consequently, we have
and
an ci
^21 C2
ci «12
C2 «22
Xi =
an ai2
«21 «22
and X2 =
an
«21
ci
C2
an
«21
«12
«22

Section 18.6 Solving Systems of Linear Equations
669
Coefficient Determinant
The denominator of these expressions is the coefficient
determinant. The numerators are based on the coefficient
determinant, with the column representing the sought
unknown replaced by the constant-term column.
This method may be extended to apply also for large systems of
n linear equations in n variables, provided that the coefficient
determinant ^ 0, as division by zero is not permissible.
To use Cramer's rule, proceed in the following manner.
o Calculate the value of the coefficient determinant:
- if it is = 0, Cramer's rule is not applicable;
- if it is ^ 0, use the determinant as the denominator of the
quotients.
o Compute the quotients.
Solve the system
2 %i + 3 %2 -
xi — 2 %2 -
5 x\ + 4 X2 -
-2x3 =
- 3 x3 = ■
-4x3 =
1 "
-9
2
>
The coefficient determinant is
D =
2 3
1 -2
5 4
2
-3
-4
= -21 * 0
consequently, Cramer's rule is applicable.
To determine xi, X2, x%, replace the first/second/third column
of D with the constant-term column of the equations:
D^ =
13-2
•9 -2 -3
2 4-4
= - 42 ; *i =
D
42
21
= 2
D2 =
2
1
5
1
-9
2
2
-3
■4
21; x2 =
D
21
21
= 1
£>3 =
2
1
5
3
2
4
1
-9
2
63; *3 =
£>3
D
-63
-21
= 3
With an increased number of system equations, Cramer's
rule leads to a rapid increase of the number of arithmetic
operations. Thus, it has no advantage over Gaussian, Gauss-
Jordan, and inverse matrix methods for solving systems of
linear equations with numerical coefficients, except when the
system consists of only two or perhaps three equations.

670
Chapter 18 MATRICES AND DETERMINANTS
On the other hand, Cramer's rule is useful for equations with
algebraic coefficients, such as
axi~ bx2 + ax% = 2b + a >
(a + b) x\ + a X2 - a x% = 0 ^
a x\ + (a + b) X2 + bx% = b
where X2, for instance, can be expressed as
X2 =
a
a + b
a
a
a + b
b
2b + a
0
b
-b
a
a + b
a
—a
b
a
—a
b
Charrxpou^ne Music
"#
An' a'nozu, folios, a lettle Cramer valtz. Turn on the bubble
machine1. !An' a 1, an' a 2, an' a 3, an' a 1-3-2, an' a 2-1-3, an' a
2-3-1, an' a 3-1-2, an' a 3-2-1 ...

671
Chapter
19
EMBARKING ON CALCULUS
Page
19.1 What Is Calculus? 673
19.2 History 674

672
r^t&- ■_ - *?-£
The scope of calculus is oceanic.

673
19.1 What Is Calculus?
Calculus (plural: calculi) is a diminutive form of Latin calx,
which means "stone", and originally derives from Greek
yoik\<5 (chalis), "limestone" (cf. English chalk, which we use
for writing on the blackboard).
Calculi, in the form of pebbles or beads, were used when
reckoning on the countingboard or the abacus - hence the Latin
verb calculare meaning "to calculate".
The word calculus became a common designation to denote all
branches of mathematics:
We say the Arithmetical Calculus, the Algebraical Calculus,
the differential Calculus, the 'Exponential Calculus, the
Jlu^ional Calculus, the Integral Calculus, the Literal or
Symbolical Calculus...
Charles Hutton, A Mothemoticol and Philosophical Dictionary (1796)
Infinitesimals — quantities smaller than any assignable
finite quantity but yet not zero - held a prominent place in the
early development of the field of mathematics that concerns
differentiation and integration of functions; hence the name
"infinitesimal calculus".
Replaced today by the concept of limit values, the word
"infinitesimal" is no longer part of modern mathematics
terminology and "infinitesimal calculus" has become "the
calculus" or just "calculus", encompassing
o differential calculus, which deals with derivatives, that is,
instantaneous rates of change of continuous functions;
o integral calculus, which is the reverse process: finding
functions when their derivatives are known;
o differential equations, which contain derivatives; and
o calculus of variations, which is the study of maxima and
minima of functions whose values depend upon a curve
or another function; the basic problem is to find a function
for which a certain integral assumes a maximum or a
minimum value.
Was Calculus invented or discovered
- Was it made or was it uncovered?
Some only deride,
Others try to decide.
Still we aren't sure if it's one or the other!

674
Chapter 19 EMBARKING ON CALCULUS
19.2 History
PYTHAGORAS (b. c. 582 B.C.)
ZENON (c. 500 B.C.)
Method of Exhaustion
PLATON (429 - 347 B.C.)
EUDOXOS (c. 370 B.C.)
EVCLEIDIS (c. 300 B.C.)
ARCHIMEDES (287 - 212 B.C.)
PAPPOS (A.D. c. 400)
KEPLER Johannes
(1571-1630)
GALILEI Galileo
(1564-1642)
TORRICELLI Evangelista
(1608-1647)
CAVALIERI Bonaventura
(1598-1647) cf.p. 446
Infinitesimals were of vital importance to the followers of
Pythagoras and to the work of Zeno of Elea. The schools of
Plato and of Eudoxus developed the method of exhaustion, in
which a sought area or volume is confined between two known,
or calculable, quantities, one decreasing and one increasing
towards the unknown quantity.
If a circle is enclosed between two /i-gons, one circum-scribed
and one inscribed, with n assuming forever-increasing
values, the difference in perimeter and in area of the two polygons
becomes successively smaller, until it approaches zero; the
perimeter or area of the circle between the polygons can thus be
determined with an accuracy that depends only on the
magnitude of n.
Calculation methods of this kind were used by Euclid, by
Archimedes, and by Pappus of Alexandria. Archimedes
employed them to determine the areas of closed plane figures, the
volumes of solids, and the locations of centers of gravity - in a
way closely resembling the integral calculus of today.
After Pappus, progress in calculus stalled until Kepler,
Galileo, and Torricelli applied the results of Archimedes to
problems in astronomy and physics. Kepler extended the use of
infinitesimals to solving problems of maxima and minima,
marking the advent of differential and integral calculus.
Bonaventura Cavalieri, a countryman and disciple of Galileo,
wrote the first textbook on integration methods, Geometria
indivisibilibus continuorum, first published in 1635.
GEOMETRIA
INDIVISIBILIBVS
CONT1NVOR.VM
Noua <juadam tationc nromotsu
A r T H a 2. X.
P. BONAVENTVRA CAVALERTO
MEDIOJ-ANEK
Qr&rtit S.HitrotLQlim in Alms BojjQijicn^rircfjigjTn*
Prtm. Ad4thtn**iiternm Frofijf.
la bacpoflrraa cdJaion* ib aroribu* (ipwgati.
Mllbtjtrtfi.D.IX
MARTIVM VRSINVM
PEHUi MAfLCHIONEM&c
4g+ CEOMETftljE
THEOREM* h PRGPGS, I.
Figurephn*quffcunqi ?neifdem paraJIdii conflitnte,
mquibusjdu^isquitjLifcunqicifcJcmparalkli^cquL
dUbuiibus re ft is I in ds* concepts? cuiufcumqj refts linex
porcioncsttmt£qui]«j«Lam inter fe aquafes eruntJ Et
figure folidff quaccumq* in cifdcm phmjp^wllclijconfti-
CutXiinquibuSjdu&s quibufcunqi plants etfdempliMj
paralMis iquiiiiftirmbus 3 concept* cuiufrunq; fie du&l
pUm in ipfo fa I id is fig ur* planae func ^quaks, parircr in-
terfe fqmlcs crum. Dicintur aurem figure xqualitcr
anilogE, fumpJinij cum ipisfQlid^mttrfe comparaise*
acccijm iuxtareguUslmeas, feu pUaa paraIJ=Ja r in qui-
busefreiuppoiuuitur, cum hociuericopuscxplicare*
LIBER VI1,
9 O N Q M T .€, M,DC,LIH.

Section 19.2 History
675
de ROBERVAL Gilles Personne
(1602-1675)
DESCARTES Rene
(1596 -1650)
de FERMAT Pierre
(1608 -1665)
WALLIS John
(1616-1703)
BARROW Isaac
(1630-1677)
PASCAL Blaise
(1623-1662)
The French mathematician Gilles Personne de Roberval
determined the area of the cycloid and of the parabola. Most
important was his description of curves as moving points;
thus, a tangent was defined by determining the instantaneous
direction of the moving point at any position on the curve.
A further step in the tangent problem was taken by the French
mathematician and philosopher Rene Descartes, who applied
infinitesimal calculus to find the tangent to a curve at a given
point on the curve.
The French lawyer Pierre de Fermat, "the king of amateurs of
mathematics", solved problems for finding maxima and
minima with methods at present used in differential calculus,
and he used integral calculus to find areas, the lengths of arcs,
the locations of centers of gravity, etc., but he did not reduce his
procedures to rule-of-thumb methods or point out that
integration and differentiation are inverse operations.
The new methods were employed to even greater purpose by the
English mathematicians John Wallis and Isaac Barrow, the
latter also a classical linguist, and the French
mathematician-physicist-philosopher Blaise Pascal, all in the mid-17th
century. Wallis dealt with infinite series and with products of
an infinite number of factors; Pascal introduced multiple
integrals; Barrow determined tangents by a method similar
to differential calculus and was also the first to recognize that
integration and differentiation are inverse operations.
LECTIONES
0PT1CJE k qEOMETMCM;
lo qiiifeui
PHENOMENON OPTICORUM
Gcmiia* l*ti*mrt JnTcftigifltuc, ic csponmnur:
IT
mt^i^M«Mf«Mmp
Aufiorc 11 A A C O BaRHOWj
CoUcgii $s* Tnntttfk in Acidemia CmuA. Prxfedo*
Et SOC1LTATIS 1ZQ1A SwJilc.
OJ m upwA m yiA* W ~«*-1*. * 1*0- ***** lf& »*
WW tin JpA?'» ^ *» Tit* W4J*#i 4 Ff^*^i J*
WxJ* d Ti *J/ • n^, JktHL
I O TLB /3C/.
fU a**ttitmi J**W , & profane *tn*la if
^**mr^^*++m^*m^im

676
Chapter 19 EMBARKING ON CALCULUS
NEWTON Isaac
(1643 - 1727)
von LEIBNIZ Gottfried Wilhelm
(1646 - 1716)
By calculations with infinitesimals, mathematicians laid the
foundation for calculus in its modern form. Their work
included
o in mechanics, notably in statics, the determination of
centers of gravity, and in dynamics, the clarification and
use of the concepts of velocity, acceleration, etc;
o the theory of maxima and minima;
o the calculation of the areas of plane figures, the volumes of
solids, and the lengths of arcs;
o the tangent problem; and
o the calculation of infinite series, infinite products, periodic
continuous fractions, etc.
It only remained for mathematicians to realize the true
relationships between these tasks, and to bring order to the
methods thus far accessible to only a narrow circle of
scientists.
This last important step was taken by the English
astronomer, physicist, and mathematician Isaac Newton and the
German mathematician and philosopher Gottfried Wilhelm
von Leibniz, who are considered the true founders of today's
calculus. Newton had used his "calculus of fluxions" as early
as 1665, but did not publish his works until 1704, while Leibniz
published his early work on calculus in 1684.
ACTA
ERUDITORUM
ANNO M DC LXXXIV
publican
SERENISSIMO FRAT&UM PARI,
DN. IOHANNI
GEORGIO IV,
Electorates Sasonici Hsredi,
DN. FRIDERICO
AUGUSTO,
Ducibus Saxoni« Scc&c^kc.
PRINCIP1BUSJUVENTUTIS
diciCi.
Cam S. Cifir?* M*)tftjtu & PotttttitjTmJ £/<%
LIP SJjE.
PTQitam aptid GftGSSnJM & j. p. GLHTI75CHIUM.
Typii CHR1STOPHQRI GaNTHERL
a-m « DC Limy.
NSttOCTOMOS^VDCUBCUY. 4^
METHQDUS PRO MAXtMIS ET A&
SUnkAX1*cu^6^wVV,V^YT,ZZ;TIariitnaha*
GSvet^%YT,ii dtlp&AX abfcifl* abace,rattur x» TMgemcjfint
V^WCiYI^ZEikttJCtancnterttfpbaiTcifipoftauBiC^p,^
Jimrtdadiqni proaxtotio aJTomuTOttrurdx) Actvda qu* ft ^
d»Iut»(¥d^Tely1vdi>eft«lVB(idWCfidYDi¥d2E)v6.
tcmrdr(¥tldv)vddyvddi) frrvAtfcxtiuU ipfrum* (*d ipb-
tuai^i^at y,tuti)Hupoiitiiuitulir^ulccrumulet:
SitiquioHtu^acocifti[i*lCT]td4^o^^c^du<ric»m»
• dirfifiiTWF(Jraotdioab^OCTutnjrf«YY,^&dUcuiviri*
dinMgrcfpoofaricnrra V V)crit dyggu.d» . }imAji&i*&S*t-
4x-"<l7>{.dw^dx.lAri!r^jCf^dxrx^vdv4,rdEt(tkpofi£d
ytqn^fictdyKpidFtf H?. IbttbknoeoimcftvdformuUE^
toy,Tdcompttidio0*0^11*^1^^7,3^1111»^ Kocmiamfti
&d*codrahaodQinhiKrilc&bttaAiti,rty&^
ipdctthnJM(^cngrfoidifetn<^ Notaodum ttiimnoa <bti
fan per itgidfiun a djj&ccoaili AvqiudoDc, rdJl cum
,jfc»d?yTd> * 1
11
"G.G.L." in the above excerpt is the Latin form of Leibniz's initials.

Section 19.2 History
677
It has been established that Newton had adequately described
his calculus as early as 1693, but this work, Tractatus de
quadratura curvarum, was not published until 1704, when it
appeared as an appendix to Opticks.
pPTICKS-
O R, A
TREATISE
OF THE
REFLEXIONS, REFRACTIONS,
INFLEXIONS and COLOURS
O F
LIGHT.
ALSO
Two TREATISES
O F TH E
SPECIES and MAGNITUDE
..OF
Curvilinear Figures.
LONDON,
Printed for Sam. Smith. andBeNr* Waliord,
Printers to the Royal Society , at the Privet's Arms xa\
St. P*«Ts Church-yard. MDCC1V-
I
n a * „
[I7i]
z, y, x, v, Sc hse ut fluxiones aliarum z, y, x, v, 8c
hec ut fluxiones aliarum z, y, x, v. Defignant igitur
z, z, z, z, z, z, z, z \2c. feriem quantitatum quarum
qucelibet pofterior eft rluxio praecedentis 8c quaelibet
prior eft tiuens quantitas fluxionem habens fubfe-
- i
quentem. Similis eft feries A'az—zz, A'az—zz,
"^■^™"—' ^»^^»,^«^™» ^B^^M»,l^H^MM i^M^^M,s#M^^^^
t/zz—zz , A'az—zz , f'az—zz , A'az—zz, ut Sc
z* az-^-z* az-(-za az^-z2 az-j-z2
fenes :»
a—z a—z
-z
a—z
az-i-z*
,. Et notandum eft quod quantitas quaelibet
prior in his feriebus eft ut area figune curviliniae
cujus ordinatim applicata reSanguk eft quantitas
pofterior & abfcifla eft z : uti A'az—zz area curvas
cujus ordinata eft ^az—zz 8c abfcifla z. Quo au-
tern fpecbnt hsec omnia patebit in Propofitionibus
qua? fequuntur.
[ijo]
TRACTATUS
D E
Quadratura Curvarum.
QUantitates indeterminatas ut motu perpetua
. crefcer.tes vel decrefcentes, id eft ut fluen-
tes vel defluentes in fequentibus confidero,defispoqj.
liceris z, y, x^v, & earum fluxiones feu celentates
• • • •
crefcendi noto iifdem Uteris pun&atis z, y, x, v.
Sunt Sc harum fluxionum fluxiones feu mutationes
magis aut minus celeres quas ipfarum z, y, x, v
fluxiones fecundas nominare licet 8c fie dignare
z, y, x, v, 8c harum fluxiones primas feu ipfarum
z, y, x, v fluxiones tertias fie z, y, x, v, Sc quartas fie
z, y, x, v. Et quernadmodum z, y, x, v funt
fluxiones quantitarum z, y, x, v, Sc has funt fluxiones
• • • •
quantitatum z, y, x, v Sc hse funt fluxiones
quantitatum primarum z, y,x, v : fie hce quantitates confide-
rari poflunt ut fluxiones aliarum quas fie defignabo,
TABULA
Curvarum JwipUciorum qua quadrari pojfunt.
Curvarum forma?. Curvarum ares.
Forma prima.
dzrl = y.
Forma fecunda.
dz*" __
ee-\-2car\-int Y'
Forma tenia.
S* = r.
="t.
i.dz.?\/rrf?=y. ^fR, = tj exiftente R~ye-^fz'
a. dz:? \Ze-\-fz-=y. "^ar* dRJ = t.
3. dz.;t- \Ze-\-fz.=y. ^ dR' = t.
4.. dz* Ve -H*=y. 5^ dR* _= t.
Forma quarta.
dz""1
1. = y.
d£*"1
= y-
-T-2ft,
31"
dR=t.

678
Chapter 19 EMBARKING ON CALCULUS
Expanding on his calculus, Newton in 1671 wrote Methodus
fluxionum et serierum infinitorum. This work was published
posthumously in an English translation (1736); the Latin
original did not appear until about 50 years after Newton's
death.
THE
METHOD of FLUXIONS
AND
INFINITE SERIES;
WITH ITI
Application to the Geometry of Curve-lines.
By the Imvixto*
Sir ISAAC NEWTON, K<-
Late Prcfideot of the Royal Society.
tranJUttdjnm the AUTHOR'i Latin Original.
mU /it -m*U ftiktuk*
T* wtkk It fifcjiirt,
A PlftfETUAL COUUINT UpOQ the whole Wpflt,
Am ho tat ions, Illditiatiomi, tiui SorntMtMTi,
Acompkat Infihution for the ufe of Learners,
By yOHX COIiO//,M.A. andF.R.8.
Mailer of Sir 7*«p* MtiUmfrtx fee Marhrmiricii-Schoot « Rtttytr,
LONDON;
Printed by Hbmry Woootalli
AodSoldby Johm No»iu, « the LW without Tmp^-JUr.
M.DCCXXXVL
P R O B._ III.
7S determine the Maxima- and Minima of Quantities.
i. When a Quantity is the greateft or the leaft that it can be,
at that moment it neither flows backwards or forwards. For if it
flows forwards, or increafes* that proves it was leis, and will pre-
fently be greater than it is. And the contrary if it flows backwards,
or decreases. Wherefore find its Fluxion, by Prob. i. and fuppofc
it to be nothing.
2. Examp. I.. If in the Equation x1 — <7#* -f- axy —y* = o the
greateft Value o£ x be required; find the Relation of the Fluxions
of x and y, and you will have 3x1:*— zaxx -f- axy —■ 3^* -f- ayx
= 0. Then making x ^.o, there will remain — 3^* ^f- ayx=oy
or 3jf* ^ ax. By the help of this you may exterminate either x
or y out of the primary Equation, and by the refulting Equation you
may determine the other, and then both of them by — 37* -{-
ax
0.
3. This Operation is the lame, as if you had multiply'd the
Terms of the propofed Equation by the number of the Dimentions
of the other flowing Quantity/. From whence we may derive the
famous
and Infinite Series;
45
famous Rule of Huddenius, that, in order to obtain the greateft or
leaft Relate Quantity, the Equation muft be difpofed according to
the Dimensions of the Correlate Quantity, and then the Terms are
to be multiply'd by any Arithmetical Progreflion. But fince neither
this Rule, nor any other that I know yet publiflied, extends to
Equations affected with furd Quantities, without a previous Reduction j
I fhall give the fbHowing Example for that purpofe.
4. Examp. 2. If the greateft Quantity y in the Equation x* —
ay%"+" ^+ "~ xx >/#y-i-xx= 0 be to be determin'd, fcek the
Fluxions of x and/, and there will arife the Equation %xx%—zayy-k-
*£* + & ^ + 6„.+ y,> ^j finc<; by ruppofltion • 0>
omit the Terms multiply'd by yy (which, to ftiorten the labour,
might have been done before, in the Operation,) and divide the reft
by xxy and there will remain 3*—*^"V^ =0, When the Re-
du&ion is made,, there will arife ^ay~\~ $xx = o, by help of which
you may exterminate either of the quantities x or y out of the pro-
pos'd Equation, and then from the refulting Equation, which will
be Cubical, you may extract the Value of the other.
HUYGENS Christiaan
(1629 -1695)
Historians have established that Leibniz and Newton both
developed the new form of calculus entirely independent of
each other - although Leibniz had doubtless heard of Newton's
works on calculus during his first visit to London in 1673.
During his years as a diplomatic counselor in Paris (1672-
76), Leibniz developed many of his mathematical ideas -
including calculus - influenced by the Dutch mathematician,
astronomer, and physicist Christiaan Huygens.

Section 19.2 History
679
BERNOULLI Jakob (Jacques)
(1654-1705)
BERNOULLI Johann (Jean)
(1667-1748)
de L'HOSPITAL
Guillaume Francois
(1661 -1704)
In any event, we are indebted to Leibniz for presenting the new
calculus in a form accessible to a greater number of readers,
while Newton's calculus of fluxions remained intelligible
only to a select few. The symbols which are still used today in
differential and integral calculus derive mainly from
Leibniz, who was also the first to realize the importance of the
new calculating methods to the entire field of mathematics.
Subsequent development of calculus was largely the work of
the Swiss brothers Jakob and Johann Bernoulli, champions of
Leibniz in his dispute with Newton over precedence to the
invention of calculus. The Bernoulli brothers were the first to
give public lectures on calculus.
The first printed textbook on calculus, Analyse des infiniment
petits, was published in Paris in 1696. No author is mentioned
in the first edition, but the second and following editions give
the author as Guillaume Francis de L'Hospital, a French
mathematician. In 1691-92, L'Hospital was instructed in
differential and integral calculus by Johann Bernoulli;
correspondence between them suggests that much of the
Analyse was the work of Bernoulli. The book, which deals
only with differential calculus, was largely instrumental in
spreading knowledge of differential calculus during the latter
half of the 18th century.
ANALYS E
DES
INFINIMENT PETITS,
four linttUigmtt des lignts emrbti*
A PARIS,
DE i'lMPRIMERIE ROYALE.
M. DC. X C V I.
Section.!
Sect. II.
Sect. IH,
Sect. IV.
Sect.V.
Sect. VI-
S£CT.VII.
Sect.VUL
Sect. IX.
Sect.X.
TAB L E.
Xy'U ton danne lesRcgfes da coitaldes
Differences; pag, u
Tjfige du csJcul des differences pour trott-
ver let Tungcntes de twtes fortes de.
lignes eoaibes > Ji,
IJf&vt da eatcul des differences poar fro*-
<ver its plas grandes ey lei moindrei
Appltquees, oufe redutfim tes qitcffhijs
De mwimis & minimis., 41.
*Vfigt da calcal des differences pour trott^
<vtr let points d'inffexhn gr de rehroaf-
fiment, jj. .
'UfAre da c&lcul des differences poar troa-
'ver let DevrfoPees., 71,
*Ufagt da cdkal des differences pour rrott*
ijtr let Caufiitjites par reflexion j 104.
IJjage du calcul des aijfertnces pour trott-
'ver Its Confronts par refft^ion, 110,
*Vfigc da cdkal des d:ffo*nces tour trott-
ver Us points des Ihnes eourhes aai
touchent une infinite de tignes donnbes
de poftrianf dtoites on courts f 131,
Solution de tjuel^ues Prahlcmts qui
dependent des Methodes prttedentef j 145*
Noxvell? maniert de ftfervir da Cahul des
differences darts Us courses vcamctri-
ques, d'ou f'on deduh U Mrthedt it
^"Dcfcartcs ^HluUc, ^4,

680
Chapter 19 EMBARKING ON CALCULUS
AGNESI Maria Gaetana
(1718 -1799)
EULER Leonhard
(1707 - 1783)
LAGRANGE Joseph Louis
(1736-1813)
LAPLACE Pierre Simon
(1749-1827)
CAUCHY Augustin Louis
(1789-1857)
p. 466
The first comprehensive textbook dealing with both
differential and integral calculus was the Italian mathematician
Maria Agnesi's Instituzione analitiche (1748), with a French
translation in 1755 and an English translation (Analytical
Institutions) in 1801. Agnesi's great learning and her
proficiency in languages brought forth her remarkable
compilation of the works of many authors.
The Swiss mathematician Leonhard Euler, the French
mathematicians-astronomers-physicists Joseph Louis Lagrange
(Italian born) and Pierre Simon Laplace, and the French
mathematician Augustin Louis Cauchy were prominent
figures in laying the foundation of differential equations.
In Institutiones calculi differ entialis (Saint Petersburg, 1755)
Euler showed how to differentiate the transcendental
functions he had described in 1748 in his Introductio in analysin.
INSTITUTIONES
CALCULI
DIFFERENTIALS
cux iiui vru
IN ANALYST FINITORUM
AC
INSTITVTIONVM
CALCVLIINTEGRALK
VOLVMEN PR1MVM
IN QVO METHODVS INTEGRANDI A PRIM1S FFIN*
■CtMIS VSQVE At) INTEGJUTtQNEli JiEQVATtOWM DlFfE*
FENTtALLVM t*l\\\ GRAQVS PERTItACTATVL
DOCTRINA SERIEELUM
»»tT4ll
LEONHARDO EULERO
ktAo. lit iciiht. it jug. irTT. jaiui*. prmicTOii
LEONHA&DO EVLERO
ACAD. SCJENT. tORVSSIAE D1RECTOHE VICENNALl ET SOCIO
ACAD. PETROP. PARISH- ET LOND1NL
■>•
••ct«*
1 h r i v ■ i ■
ACADCMIAE IMPERIAL!* SCIENTIARUM
MTtOfOlITAHAI
I T t 1.
PETRQPOLt
IropeoG* Acidcmije Impcriilii Scienterdm
Greek, isos, "equal"
ZENODOROS
(c. 180 B.C.)
Euler's Institutiones calculi integralis (3 volumes, Saint
Petersburg) was published in 1768 - 70. For almost a hundred
years the Introductio... and the Institutiones... were the
dominating textbooks in higher mathematical education.
Isoperimetric problems deal not only with isoperimetric
curves, that is, closed curves with equal perimeters in the
plane, but also with closed surfaces in space. The study of such
problems emerged in antiquity; Zenodorus proposed in On
Isometric Figures that the circle has the maximum area of all
isoperimetric figures in a plane, and that the sphere has the
maximum volume of all bodies with equal surface. Euler

Section 19.2 History
681
FOURIER Jean Baptiste Joseph
(1768-1830)
GERMAIN Sophie
(1776 -1831)
ABEL Niels Henrik
(1802 -1829)
developed in Methodus inviendi a general method to find a
function for which a given integral assumes a maximum or
minimum value, thereby inducing isoperimetric problems to
become a separate mathematical discipline, later known as
calculus of variations.
METHODUS
I N V E N I E X D I
L1NEAS CURVAS
Majimi Mmimivc propriciarc gaudemei,
S I V E
SOLUTIO
PKOBLtMATIS ISOPERIMETRIC!
LiTliWMO iKNSU iCCEtTl
A U C T O Jt £
LEONHARDO EULERO.
PraftfoM %», tf Acadtmit IififtrH&t StitntU-
T1UB PETlLGFOniAffJE Jctlfl.
ar > ^i
UUiANNi li OENEVi,
A?1|J MuCUHMrCHULIU Boutty/ET tt SosJm.
W O C C X L I V.
Leonhard Euler's A Method for Discovering Curved Lines Having a
Maximum or Minimum Property or the Solution of the Isoperimetric
Problem taken in its Widest Sense (1744).
The French engineer-mathematician, Egyptologist, and one
of Napoleon's most able administrators Joseph Fourier
presented, in 1828, in his Theorie analytique de la chaleur
("The Analytical Theory of Heat") a method to express any
function as the sum of fundamental trigonometric functions.
Fourier's discovery is of fundamental importance in both
physics and mathematics, where it revolutionized the solving
of partial differential equations (that is, differential equations
involving more than one independent variable).
Although higher education for women was virtually
nonexistent, and despite prevailing prejudice and skepticism
about feminine pursuits in fields of scientific research, the
18th and 19th centuries brought forth extraordinary
achievements by strong-minded, scholarly women. In mathematics,
the Frenchwoman Sophie Germain - one of the founders of
modern applied mathematics - rose to fame in 1816 for a paper
on calculus of variations applied to the theory of elasticity.
Struggling against poverty and illness, the Norwegian
mathematician Niels Henrik Abel made significant contributions
to several fields of mathematics during his short life. In
calculus he will be especially remembered for his
contributions to the theory of integral equations, that is, equations in
which the unknown function belongs to an integral.

682
Chapter 19 EMBARKING ON CALCULUS
CAUCHY Augustin Louis
(1789-1857)
WEIERSTRASS
Karl Theodor Wilhelm
(1815-1897)
KOVALEVSKI Sonia
(1850-1891)
LIE Marius Sophus
(1842 -1899)
Galois theory: p. 301
Lie Groups
POINCARE Jules Henri
(1854 -1912)
FREDHOLM Erik Ivar
(1866 -1927)
SCHWARTZ Laurent
(b. 1915)
In the second half of the 19th century infinitesimals were
replaced by the concept of limits, a development due mainly to
the work of the French mathematician A. L. Cauchy and the
German mathematician Karl Weierstrass.
A student of Weierstrass's, the Russian mathematician and
novelist Sonia Kovalevski made important contributions to
integral calculus and differential equations. In spite of her
academic qualifications and letters of recommendation from
the otherwise powerful Karl Weierstrass, she was for a long
time unable to obtain an academic position; no university
would accept a woman mathematician, however qualified. In
1889 she was appointed professor of mathematics in Stockholm,
but many still resented the appointment of a woman, among
them the Swedish playwright and novelist August Strindberg.
The Norwegian mathematician Sophus Lie developed a
geometric integration theory for certain partial differential
equations. Lie is best known for his expansion of group theory,
initiated by Galois in the theory of algebraic equations, to
include groups of geometric transformation, today known as
Lie groups and exerting great influence on research in the
theory of differential equations and in other fields of
mathematics and its applications, especially in quantum physics.
The French mathematician, physicist, and astronomer Henri
Poincare, "the last mathematical universalist", pioneered
work on the theory of differential equations and its
applications to dynamics and astronomy. He published over 500
articles and books dealing with practically all the major
fields of pure and applied mathematics; his publications on
differential equations span from his thesis in 1878 to his last
paper, in 1912.
The Swedish mathematician and physicist Ivar Fredholm
proved the existence of solutions for a general class of integral
equations, which is important for the study of boundary
problems in mathematical physics. This contribution,
sometimes characterized as "long overdue", gave significant
inspiration to the development of functional analysis; in a
modernized form, Fredholm's ideas are still of fundamental
importance in the theory of linear partial differential equations.
Before the 1950s the study of partial differential equations
was mainly devoted to elliptic, parabolic, and hyperbolic
differential equations originating in physics and
representing phenomena such as electrostatics, heat conduction,
and electromagnetic wave propagation. The introduction of
distribution theory by the French mathematician Laurent
Schwartz at the end of the 1940s made it possible to develop a
theory of general linear partial differential equations, first
with constant coefficients. The extension to equations with
variable coefficients led to the development of a theory of
pseudo-differential and Fourier integral operators,
incorporating formal methods used in mathematical physics. This
forms the core of microlocal analysis, which has also turned
out to be an effective tool in the study of nonlinear partial
differential equations.

683
Chapter
20
INTRODUCTION
TO DIFFERENTIAL CALCULUS
Page
20.1 Derivatives and Differentials 685
20.2 Differentiating Algebraic Functions 689
20.3 Differentiating Transcendental Functions 695
20.4 Special Techniques of Differentiation 705
20.5 Partial Differentiation 710
20.6 Mean-Value Theorems 716

684
-^^s^/^i^-
.-«5?

685
20.1 Derivatives and Differentials
If the independent variable x of a function f(x) is increased by
a small amount Ax - the increment "delta x" - it will cause a
corresponding change Af of fix); the ratio Af/Ax is a
measure of the rate of change of/"with x.
fix)
fix + Ax)
x
fix)
fix + Ax)
fix + Ax) -f ix)
fix + A x) - fix)
x4
If Ax is made progressively smaller, the limit value
,. fix+ Ax)-f jx) _ Af
lim = hm
Ax ->0 A X
A x
Ax —> 0
when Ax tends to zero (if it exists) is the first derivative of fix)
with respect to x,
df
d x
= f (*); f = T prime":
which describes the instantaneous change of f at a given
point x.
If the limit value /*' (x) exists in a specific interval of x, the
function /*(x) is said to be differentiable in this interval.
Differentiation of f \x) gives the second derivative f"(x) -
read "f bis" - and, analogously, the derivative of/*11 is the
third derivative f' '(#); higher derivatives are denoted fi v ix)
for the fourth derivative, fv for the fifth derivative, etc.
Find the first and second derivatives of fix) = x2 .
A x
= ix + A x)2
= x2 + 2xAx + (Ax)2
= 2 x A x + (A jc)2
= 2x + Ax
— xl
fix) = lim i2x + Ax)
A x -^ 0
= 2jc.
T(*)
fOc + A jc)
f\x + Ax)-fix)
fix + Ax)-f\x)
A x
r\x)
= 2x
= 2 ix + A x)
- 2x+2 Ax -
2 x + 2 Ax
A x
= lim 2
A x -^ 0
-2jc
-2x
= 2.

686
Chapter 20 INTRODUCTION TO DIFFERENTIAL CALCULUS
The notation
von LEIBNIZ Gottfried Wilhelm
(1646 - 1716)
LAGRANGE Joseph Louis
(1736-1813)
ARBOGAST
Louis-Francois-Antoine
(1759 -1803)
T H £ O R I E
DES FONCTIONS ANALYTIQUES,
CtMIIl.1T
LIS PRINCIPES DU CALCUL DIFF^RENTIEL,
D'tMriXIMINT tlTITS OU D'iVAMOUlSSANS.
DC LIXtTCS OU PC FLUXIONS.
B-T l(l' 1"
A L* ANALYSE ALGEBRIQUE
Ptf J. i. IAC&AHCI, 4* rioiuiui cuiioiuL
A PARIS.
PI I'UMUUIII Dl LA ltruuiQuc.
Praiiial an V.
dnf(x)
dxn
for the n-th derivative of f(x) was introduced by Leibniz.
In his treatises Theorie des fonctions analytiques (1797)
and Legons sur le calcul des fonctions (1800), Lagrange used
Leibniz's notation, and introduced his own notation
The French mathematician and lawyer Louis Arbogast
introduced in his Du calcul des derivations (1800) the notation
Dnf(x) or Dnf
for the /i-th derivative of f.
We have so far regarded df/dx and dy I dx as inseparable
symbols for the derivative of a function y = f(x), but we may
consider dy / dx as the ratio of two separate quantities dy and
dx, known as differentials, which can be handled just as any
other algebraic quantities, as we shall see in the chapters on
integral calculus and differential equations (Chapters 21 and
34).
The publishing date Prairial an V
refers to the 9th month (20 May -
18 June) of the year five (1797) of
the French republican calendar.
Derivatives of Linear Functions
A linear function, that is, a function of degree one, is
represented geometrically by a straight line with the slope
m =
Ay
A x
y2-yi
X2 ~ Xi
Xi ^ X2
X
which is also the derivative y' of the linear function y =f(x);
all higher derivatives are zero.
In the special case of y = constant, the line is horizontal with
the slope = zero; consequently, the derivative of a constant is
zero.

Section 20.1 Derivatives and Differentials
687
Derivatives of Nonlinear Functions
Functions of the second and higher degrees are represented
geometrically by curves of various descriptions.
fXxo+Ax)
f(xo) ~
secant
tangent
x
The first derivative of a nonlinear function is the gradient of
the curve, that is, the slope of the tangent to the curve.
The slope of the secant PQ is
f(xo + A x) -f(x0)
m
PQ
A x
As Q moves continuously closer to P, the secant approaches the
tangent, with the slope
f(xo + A x) -f(x0)
mp = lim
A*-> 0
A x
= f'(x),
which is the derivative at xq of the function fix).
Determine the first, second, and third derivatives of the
function f(x) = x3 and trace their graphs.
f(x) = x
f"(x) = 6x
f"(*) = 6
x
fix) =x
secant
x0 + Ax
x
r (x)
r * (*> =
lim
A*-> 0
(x + A x)3 - x3 _ ^2
A x
= 3x<
3 (x + A x)2 - 3 x2 „
lim ■ = ox
Ax-> 0
A x
f" (*) =
.. 6 (x + A jc) - 6 x
lim =6
A^:^ 0
A x

688
Chapter 20 INTRODUCTION TO DIFFERENTIAL CALCULUS
The equation of the tangent at a specific point on the graph of a
function may be determined when the derivative of the
function is known.
The derivative of y = x2 is 2 x. At x = 1.5 ; y = 2.25, the slope of
the tangent is
2 • 1.5 = 3
and, thus, the equation of the tangent is
y - 2-25 _ „
x - 1.5
y = 3x-2.25
and
12x-4y = 9
The slope angle a of the
tangent is given by
tan a = 3 ; a & 72c
.5, 2.25)
a
i i 1 -H
12 3 4 5
-2
x
Find the equations of the tangent and the normal of the curve
y = 3x2 at (1,3).
fix) = 3x2
— tangent
normal
x
The slope of the tangent is y' = 6 x, at (1, 3) it is y' (1) = 6 ; the
equation of the tangent is then
y = 6x + 6;
since x = 1, y = 3, the equation of the tangent is
6x-y = 3 .
The normal is perpendicular to the tangent and thus has the
equation
1
y = ~6* + a;
with (1, 3) inserted, the equation of the normal is
x + 6y = 19 .

689
20.2 Differentiating Algebraic Functions
The derivative is usually found by proceeding from already
known derivatives of the most common functions and
applying rules for differentiation of sums, products, and
quotients.
The Derivative of a Power of a Variable
d . n. _. (x + A x)n - (x)n
[xn] = 11 m
A x —> 0
xn + nxn~1(Ax) + Q) xn~2 (A x)2 + ... + (A x)n - xn
Binomial Theorem : p. 140 = 1 i m —
A x —> 0
= nxn~1 + 0+...+ 0
= n xn ~ * ,
which applies for all real numbers n.
f(x) = x4 => fy{x) = 4*4-1 = 4 x3
g (x) = xn => g ' (x) = nx71'1
-1 - r-473 -1
h(x) = x-1/3 => h'(x) =^-^-^-1 = -
3 33 x4/3
F (x) = xe => F% (x) = ex?'1
G (x) = x1 => G' (*) = x1"1 = x° = 1
#(*)=*° =» #'(*) = Ox0"1 = 0 •
If a function includes a constant factor c, it is reproduced in the
derivative,
■^lcf(x)] =cjW.
Find the fourth derivative:
• g(x) = 2x1/2
1/2
_1 _ r-3/2
2 2 2*3/2
3_
5/2
-15
7/2 *

690 Chapter 20 INTRODUCTION TO DIFFERENTIAL CALCULUS
h(x) = 4 X~1/3
h'(x) = *(jf)
—4 -4
^173-1 _ _ --43 _ *
3 " 3x4/3
16
9xm
-112
/,'" / x _ i£ /zZ.^ r-7/3- 1 - ~112 r-10/3 _
, . , , -112 /-10\ in/q . 1120 iq/q
J, iv /v\ I I --10/3 - 1 v-13/3 _
11 w ~ 27 \ 3 / "81
27jc10/3
1120
81jc
13/3
The Derivative of a Sum
The limit value of a sum of functions is the sum of their
several limits: thus if
and
we have
h (x) = f{x) +g(x)
-,,, ... fix + A x) - f(x)
f ix) = lim
Ax->0 A X
w v ,. g ix + Ax) - g jx)
g'ix) = hm
A*->0 ax
h'ix)= ff(x)+g'(x) .
This equality does, of course, apply for a sum of any finite
number of functions.
The subtraction of a function is equivalent to the addition of its
corresponding negative function. The derivative of
hix) = fix)-g ix)
is
/i'(x)= f*ix)-g*ix) .
The Derivative of a Product
The derivative of the product function
fix) - g ix)
is
n/\ t f(x + Ax)gjx + A x) - fix) g jx)
h ix) = lim
A*->0 A x
where we add and subtract the expression/"(jc + Ax)g ix) in the
numerator and obtain, after rearranging,
/i' ix) = lim
Ax -^ 0
fix + Ax) - fix) gjx + A x) - gjx)
g ix) + fix + A x)
A x Ax
= fix) • gix) + fix) • g \x) .

Section 20.2 Differentiating Algebraic Functions 691
The formula may be generalized to apply to products of any
finite number of functions.
If one of the functions is a constant, e.g., f = const, f = 0, the
derivative simplifies to
fc'(x) = f-g'(x) .
The Derivative of a Quotient
The derivative of the quotient
hix) = —r-- ; gix) * 0
gix)
is
f(x + A x) f(x)
!..• \ t g(x + A*) gW
h \x) - lim
h ' ix) - lim
Ax-> 0
.. fix + A x) - g ix) - fix) g ix + Ax)
= lim —
A x ^ o Ax • g ix) - g ix + Ax)
which we may rewrite
fix + A x) - fix) gjx + A x) - gjx)
Ax Ax
■■- . a ix) — fix) ' ^~~~~~^"^^
^(x) g ix + Ax) gix) • gix + Ax)
and obtain, finally,
fix) . gix) -fix) . g'ix)
h'ix) =
igix)Y
Derivatives of Composite Functions
In a composite function y = if o g) ix) — "a function of a
function" - f is the external function and# the core function.
The derivative of a composite function is the product of the
derivative of the external function f with respect to the core
function g multiplied by the derivative of the core function g
with respect to the independent variable x,
provided that the functions are differentiable, /"with respect to#
and g with respect to x.
Chain Rule The formula is called the chain rule of differentiation.

692
Chapter 20 INTRODUCTION TO DIFFERENTIAL CALCULUS
General Power Rule
Differentiating a power function of the general form
y = un ,
where u is a differentiable function of a variable x, the chain
rule gives
dy dy du d
du
y = -^- = -^- . = [un ]
dx du dx du dx
and thus the general power rule,
dy
dx
= nu
n-l
du
dx
If one arrives at an answer that is of different algebraic form
than a given answer, the two answers may be compared by
substituting a constant for the variable. Unequal numerical
values definitely signify a wrong answer, whereas equal
values indicate the possibility of a correct answer.
There is no stipulation how far a derivative should be
simplified, other than that it must be written in a form suitable to
its application.
Find the second derivative of
/W = (2jc + 5)5 .
fix) = 5(2x + 5)4-2 = 10(2x + 5)4
f (x) = 4-10 (2x+ 5)3. 2 = 80(2*+ 5)3.
Find the first derivatives:
g (x) = (4 x - 3 x2)3;
g'(x) = 3(4x-3 x2)2 (4 - 6 x) = 6 (2 - 3 x) (4 x - 3 x2)2 .
fit) = t VTT72;
fit) = til + t2)y2,so
fit) = l-il + t2)y2 + t^il + t2)-y2'2t
= [(1 + ^) + ^(1+^)-172 =
l + 2f2
VT772 '
Fix) = a/ ^4
\ X + 1
F'ix) =7T
\(x- lV172 _d_
dx
2 U+ 1
y
x - 1
x + 1
1 6c + i"\
2U-1
1/2
j
jx + 1) • 1 - jx - 1) • 1
(* + 1)2
W
JC + 1
JC
1 ' (JC + 1)2 *

Section 20.2 Differentiating Algebraic Functions
693
Differentiability
One-Sided Limits:/?. 358
A function is differentiable only where it is continuous, but the
reverse is not necessarily true; a function may be continuous
at a certain point without being differentiable at that point.
The derivative may have different values
fjxp + Ax) -f(x0)
o from the right, f\xQ+) = lim
A x —> 0 +
o from the left, f \xq -) = lim
A x
fjxp + Ax) -f(x0)
A x
The function is differentiable at xq only if the derivatives from
the right and from the left are equal.
The absolute-value function f(x) - \x\ is a case in point; at
x = 0, the derivative from the right, for A x > 0, is
f\0+) = lim
A x -> 0 +
/(0 + A x) -/(0)
A x
= +1,
and the derivative from the left, for A x < 0 is
r(0_)= lim m1Ax1-fio1 = _i
Ajc-^O
A x
Consequently f(x)= \x\ is not differentiable at x — 0
•A/ J ^~ | «A/
X
In the general case fix) = | xn \ , we must distinguish two
cases when n is odd:
x > 0 ; f(x) = xn ; f (x) = n • xn ~ 1 \f '(0) = 0, only if n > 1] ;
x < 0 ; fix)= -xn\ fix) - - n • xn ~ 1 .
A function f ix) may also be continuous but not differentiable
at x = xq if
t . fixo + Ax) -fjxp)
lim : = + oo or - oo ,
Ax-^0
A x
which means that the tangent of fix) at xq is vertical.

694
Chapter 20 INTRODUCTION TO DIFFERENTIAL CALCULUS
3r-
Isf(x) = yx differentiable at jc = 0?
At jc = 0, we have
f
x = 0
\^c) = H
Vo +ajc - Vo
m
Ax->0
A jc
= lim
A x —» 0
11^
\j (Ax
)'
rr oo
6r-
Although f(x) = yx is continuous at jc = 0, it is not
differentiable as the tangent at jc = 0 is vertical.
Find the first derivative of
f(x)= |9-jc2| .
|9-jc2| = jc2-9 if (9-jc2 < 0) ; f ' (jc) = 2jc
|9-jc2| = 9-jc2 if(9-jc2>0); f*(x)=-2x
The graph of f (jc) = | 9 - jc2 | shows sharp cusps at jc = 3 and
x = -3, where the function is not differentiable:
f(x)= I 9-jc2
x
-4 -2

695
20.3 Differentiating Transcendental Functions
Fundamental Trigonometric Functions
sin x
pp. 529 - 30
dx
[sin x]
= lim
A x —> 0
= lim
A x —> 0
sin (x + A x) - sin x
A x
sin x cos A x+ cos x sin Ax- sin x
A x
sin Ax . _. 1 - cos A x
= cosx- lim —: sin x • lim —
Ax-> 0 A X Ax-> 0
= 1 • cos x - 0 • sin x = cos x .
A x
COS X
d d
j— [COSXJ = -;—
dx dx
n
sin |-
X
7C
= (-1 ) sin |—- x
= — sin x
tan x
Using the formula -t—
u
v
uy v - u v'
v
dx
[tan x] =
d
d x
sin x
cos x
cos x • cos x + sin x • sin x
cos2 x
cos2 x
= sec2 x = 1 + tan2 x .
cot x
dx
[cot x] =
_d_
dx
cos x
sin x
sin x • sin x - cos x • cos x
sin2 x
sin2 x
= - esc2 x = -(1 + cot2 x).
sec x
Using the formula -j—
r
u
u
u
2 '
d r i d
-:— [secxj = -:—
dx dx
cos x
- sin x tan x
cos2 x
cos x
= sec x • tan x
CSC X
d d
-;— [cscxj = -;—
dx dx
sin x
cos x
sin2 x
cot x
sin x
= — esc x • cot x
The first derivative of y = sin 2 x3 is
y ' = 6 x2 cos 2 x3
The first derivative of y = 5 Vsin 2 x is
i j
y ' = 5 • - (sin 2 x)"1/2 • ^- [sin 2 x]
^2 dx
1,. ^ i/o 5 cos 2x
= 5 • - (sin 2 xYm • 2 • cos 2 x = -===
^ \sin 2 x

696
Chapter 20 INTRODUCTION TO DIFFERENTIAL CALCULUS
p. 512
Find the first derivative of
f{x) = cosx2; g{x) = xcosx; h{x) = cos2x .
f\x) = (-sinx2)-2x = -2xsinx2.
g' (x) = x (- sin x) + (cos x) • 1 = cos x - x sin x .
h\x) = -2 sin xcosx (= -sin2x).
Find the second derivative of
y = cos 2 x3 .
y' = (-sin2x3)(6x2) =-6x2sin2x3;
y " = - 6 x2 • cos 2 x3 • 6 x2 - 12 x • sin 2 x3
= -12 x (3 x3 • cos 2 x3 + sin 2 x3).
Find the second derivative of
y = cos
x
2 '
1 . x
y ="2sin2
1 x
y = -- cos -.
Find the first derivative of
y
sin x - x cos x
cos x + x sin x
p. 508
(cos x + x sin x)
d x
[sin x - x cos x]
A
J
y =
(cosx + x sin x)2
(sin x - x cos x)
d x
[cos x + x sin x]
A
J
(cosx + x sin x)2
(cos x + x sin x) x sin x - (sin x — x cos x) x cos x
(cosx + x sin x)2
xi
x2(sin2x + cos2x)
(cos x + x sin x)2 (cos x + x sin x)2
Find the first derivative of
fix) - cos (cos x) ; g (x) = sin (cos x) ;
h (x) = sin (sin x) ; F (x) = sin2 (cos x) .
f ' (x) = {- sin (cos x)} (-sin x) = sin x {sin (cos x)}
g' (x) = {cos (cos x)} (-sin x) = - sin x {cos (cos x)}
hl (x) = cos x {cos (sin x)} .

Section 20.3 Differentiating Transcendental Functions
697
F1 (x) - 2 {sin (cos x)} < -t— [sin (cos x)]
= 2 {sin (cos x)} { [cos (cos x)] (-sin x) \
= -2 sin x {sin (cos x)} { cos (cos x)} .
Find the first derivative of y = tan3 4 t.
y' = 3(tan4*)2 -77 [tan 4 t] = — .
V d* ) cos2 41
x
Find the second derivative of f{x) = tan —
/"(
/'"
x)
(x)
1
" 2
1
~ 2
X
sec2 — .
d
dx
-
sec2
X
2
1 „ x ( d
= 2'2'SeC2 dT
X
sec 2
•A/ «A/ «A/ I J- 1 J. (^ «A/ «A/
= SeC2SeC2tan2 2 =2seC 2tan2
Inverse Trigonometric Functions
dx
arcsin X If y = arcsin jc , then jc = sin y ; -r— = cos y ; dx = cos y dy =
p. 508
arccos x
arctan x
arccot x
arcsec x
arccsc x
dy
so
d [arcsin x]
d x
cosy Vl-sin2
y = arccos x ; x = cos y ; d x = - sin y dy
d [arccos x] - 1 - 1
Vl -x:
*1= X
dx cos y
|jc| < 1.
d x
siny ^ l-cos2y
yl-x'
\x\ < 1.
y = arctanx ; x = tany ; dx = sec2y dy
d [arctan x] 1 1
d jc
sec2y l + tan2y 1 + x2
y = arccot x ; x = cot y ; dx = - esc2 y dy
d [arcct x] - 1 - 1
- 1
dx
esc2 y
y = arcsec jc ; x = sec y ; dx =
d [arcsec x]
1 + cot2 y 1 + x2
tany
cos y
1
dy
dx
sec y • tan y # Vx2 - 1
; |x|>l
. coty
y = arccsc x ; x = esc y ; dx = - —:
J J ' sin x
dy
d [arccsc x]
dx
- 1
- 1
esc y • cot y # Vx2 - 1
|x| > 1.
We note that although inverse trigonometric functions are
transcendental, their derivatives are algebraic functions.

698
Chapter 20 INTRODUCTION TO DIFFERENTIAL CALCULUS
The first derivative of
is
y =
i +
'*
^3
x
y = arctan —
J 3
d |~x
dx 3~
n x2
1+9"
9 + x2
The first derivative of
y = arctan (2 x2
-x)
IS
y =
1 + (2x2 - x)2 dx
[2x2-x] =
4x- 1
4x4 - 4x3 + x2 + 1
The first derivative of
y =
1 + arctan x
1 - arctan x
is
(1— arctan x) -:— [1+ arctan x] -(1+ arctan x) ^— [l
arctan x]
y =
(1 -arctan x)
f 1
V
1+x2
(1 — arctan x)2
-(1 + arctan x)
f ~ 1
V
1+x2
(1 - arctan x)2
(1 + x2) (1 - arctan x)2
At the outset of differentiation
We needed strong dedication.
9{pzu, after a breads
And some sCices of cafe
We're set for more mat/t gratification.

Section 20.3 Differentiating Transcendental Functions
699
Logarithmic and Exponential Functions
Natural Logarithms
d n i *
t— llnxj = —
dx x
Txri. o d In (x + A x) - In (x)
Why? -j— |lnx] = lim
Ax —> 0
A x
lim
A* —> 0
1 _ (x + A x
• In
A x
x
which may be rewritten
dx
[lnx] = lim —
Ax->0
X
X
A x
In
x + A x
x
1 J ( A^/Ax
-i lim In 1+ —
x
Ax^O
x y
x
where, with -—= n tending to », we recognize the definition of
e, and have with In e = 1,
d n i l
-7— llnxj = — .
ax x
Find the first derivative:
• f(x) = ln(4x2-3);
fix) =
4x2-3
8x =
8x
4x2-3
• F (x) = In (ex);
F (x) = lne + lnx = 1 + lnx;
F'(x) =- .
x
g (x) = x2 In x ;
#' (x) = x2 -p- [In x] + In x -5— [x2] = x (2 In x + 1)
dx dx
G(x) =
In x
x
G' (x) =
x • — - In x
x
~2
1 - lnx
xi
• /1 (x) = sin (lnx) ;
w t \ n ^ d n 1 CQS (ln *)
h (x) = cos(lnx)^— llnxj =
dx x

700
Chapter 20 INTRODUCTION TO DIFFERENTIAL CALCULUS
H (x) = In3 x4 ;
W (x) = 3 (In x4)2 -r- [lnx4]
dx
o ,, a,? 1 q 12 (In2 x4
= 3 (lnx4)2 —- • 4x3 =
x4 *
y = In {In (2 x3)}
y' =
In (2 x3) <*x
d D112*3] = -^^r^r6x2 =
In 2x3 2x3
x In (2 x3)
p. 510
Find the first derivative of
y = In (tan2x) .
1
y =
tan 2x
cos 2 x
• sec2 2 x • 2
= 4 esc 4x
sin2x cos22x sin 2 x • cos 2 x sin 4 x
Find the first derivative of
y = In x (arctan x) .
. 1^ 1 ., In x arctan x
y = In x — • 1 + arctan x — • 1 = + .
1+x2 x l+x2 x
Find the equation of the tangent and the normal to the curve
fix) - x In x at (e, e).
fix) = 1 + In x
The slope of the tangent at (e, e) is 1 + In e = 2; consequently,
that of the normal is — «■.
The equation of the tangent is
y - e = 2 (x - e);
and that of the normal is
y = 2x-e
y-e = --ix - e); y = -- +
x 3e
2 2*

Section 20.3 Differentiating Transcendental Functions
701
Logarithms to Any Base
p. 156
jjj Wogax] = (logae)-
Why? y = logax =
]nx
In a
1
-7— \\ogax] = -, -7— \\nx] = : •-
dx u In a ax In a x
.- = (logae)-
Differentiation is facilitated by choosing e as logarithm base.
Although logarithms are transcendental, their derivatives are
algebraic functions.
Find the first derivative of
y = log
x V* - 2
10
y = log10 * + £ loSio (* - 2) - log10 3 .
y ' = Oog10 e) - + J (log10 e) ^^
= lQgioe
1_ 1
x + 2 (x - 2)
= logioe
3 * - 4
2 x (x - 2)
Exponential Functions
To find the derivative of the exponential function
y = e*,
take the logarithm of both sides,
dy
— = djc
dy
dx
— y _ qX
Iny = x ;
Thus, we have
y = ex ; y = e* ; y" = e* ; y'" = e* , etc.,
that is, the function e* is its own derivative ad infinitum.
The derivative of the function
y = af{x)
is obtained by observing that
af(x) _ elna-f(x)
y = o^lna-jj- |/(x)]

702
Chapter 20 INTRODUCTION TO DIFFERENTIAL CALCULUS
Find the first derivative:
fix) = 62x;
fix) = 62*ln6 -^ [2x]
= 62x (In6)2 = 21n6-62*
g ix)
g% (*)
h (x)
h} (x)
Fix)
q ln7 x
Gix)
In e*
_ e-x
- e~x j— [- *3] =-3 x2 e~x .
dx
— p, O X — Zi
(e5*3~2)
_ q In 1 x .
dx
[5x3-2] = 15x2.e5*3-2
= 1 x, soF' ix) = 7
= In e x ;
= x, so G' ix) = 1 .
Hyperbolic Functions
p. 536
Since hyperbolic functions can be defined in terms of
exponential functions, their derivatives may be found by
applying the rules for differentiating exponential functions.
sinh x
cosh x
tanh x
-:— [sinhx] =
dx
■j— [coshx] =
-;— [tanhx] =
ax
d
dx
d
dx
d
dx
/~\ .X- t_^ ^ JL-
2
e* + e-*
2
sinh x
cosh x
qX _j_ Q X
= coshx
/~\ .X- t_^ ^ Jl-
= sinhx
cosh x • cosh x - sinh x • sinh x
cos
h2
cos
h2
= sec
h2 x .
coth x
dx
[coth x] =
_d_
dx
cosh jc
sinh jc
sinh2 x - cosh2 x
sinh2 x
- 1
sinh2 x
- - cosech2 x
sech x
dx
[sechx] =
_d_
dx
cosh x
- sinh x
cos
h2*
= - sech x • tanh x
cosech x
d d
-\— [cosech x] = -:—
ax ax
sinh x
- cosh x
sinh2 x
= - cosech x - coth x

Section 20.3 Differentiating Transcendental Functions
703
Find the first derivatives:
f (x) = sinh V* ;
/*' (x) = (cosh ^[xj-t— l4x\ =
g (x) = sinh (x2 + x + 4);
cosh yx
2^x
g' (x) = cosh (x2 + x + 4) -t— [x2 + x + 4]
= [cosh (x2 + x + 4)] (2 x + 1) = (2 x + 1) cosh (x2 + x + 4)
h (x)
hy (x)
In (sinh x2) ;
1 d
sinh x2 &x
cosh x2 d
sinh x2 &x
[sinh x2]
_ 9l coshx2 rt
[*2] =.- 9 -2¾
sinh #z
= 2 x coth x2.
p. 542
F (x)
= coth— ;
x
F* (x) = -\ cosech2 -1(-¾-2) = —- cosech2 — .
\ XJ x2 x
G (x) = tanhe2*;
G' (x) = (sech2e2*)e2*-2 = 2e2*seche2*.
H (x) = cosh2 (3 x2 + x + 3);
W (x) = 2cosh(3x2 + x + 3)^ [cosh(3x2 + x + 3)]
= 2 (6 x + 1) cosh (3 x2 + x + 3) sinh (3 x2 + x + 3)
= (6¾ + 1) {2 cosh (Sx2 + x + 3) sinh (3x2 + jc + 3)} .
Applying the double-angle identity
2 cosh x sinh * = sinh 2 * ,
we have
H'(x) = (6x + l)sinh2(3x2 + x + 3).
Find the second derivative of y = cosh ^fx .
y = (sinh V*)-r- [V*]
d#
sinh Vjc
2-(x
y" =
dx
sinh Vjc
2^
Differentiating the quotient, we find
2^x
'cosh yx^
y
11
v
2^x
- (sinh yx)
J
^[x
4 x
Vjc - cosh Vjc - sinh Vjc
4 x V *

704
Chapter 20 INTRODUCTION TO DIFFERENTIAL CALCULUS
Inverse Hyperbolic Functions
p. 545
arsinh x
Inverse hyperbolic functions are logarithmic and, since the
derivatives of logarithmic functions are algebraic functions,
we may expect the derivatives of inverse hyperbolic functions
to be algebraic functions.
dx
[arsinh x] = -r— lln (jc + Vx2 + 1) I =
1 +
x +
x
Vx2 + 1
V*2 +1
V*2 +1
The derivatives of the remaining area hyperbolic functions
are found in a similar manner.
arcosh x
artanh x
arcoth x
arsech x
arcsch x
dx
d_
dx
d_
dx
d_
dx
d_
dx
[arcosh x]
[artanh x]
[arcoth x]
[arsech x]
[arcosech x] =
l
1-x2 ;
1
1-x2 ''
1
x Vl -x2
1
\x I ^1 + X2
\x\ >1
|*| <1
|*| > 1
; 0 <x< 1
x * 0 .
Find the first derivatives:
fix) = arsinh 2 x ;
fix) = 2
gix) =
g'ix) =
hix) =
V4r2 + 1
artanh e x :
1
e* =
qX
1 — e2x 1 - e2 x
arcosh I ;
X
/i' ix) =
V
^2-i
i- 1 *-2) =
V
x
V5
2
- 1
V
X'
(V^ = |*|)
^- 1
V
l-*2
V
JCi
X'
x
VT^2
i X
- X
x2 Vl -x2

705
20.4 Special Techniques of Differentiation
Logarithmic Differentiation
A function is often more easily differentiated if first
transformed into logarithmic form; this might considerably reduce
the algebraic labor involved in differentiating products and
quotients, and in handling composite functions.
Logarithmic differentiation is normal procedure for functions
where the variable occurs both as a base and as an exponent.
• y = xx ; determine yx by logarithmic differentiation.
Take the natural logarithm of both sides,
In y = lnxx
= x In x .
Differentiate both sides with respect to x :
1 dy 1
— • -:— = x • — + Injc = 1 + In jc
y ax x
dy
where we replace y by x x,
-— = xx (1 + In x).
ax
• y = xx*; determiney1 by logarithmic differentiation.
Take the natural logarithm of both sides,
In y = In x x
= x x In x .
Differentiating both sides with respect to x gives
- xx • — + In x \x x (1 + In x)\
x
= xx~1 + lnx {xx (1 + In x)} .
Solve for dy/dx,
— = y [xx " 1 + xx In x (1 + In x)},
and replace y by x x ,
-— = x** [xx~ 1 + xx In x (1 + In jc)}
dx l

706
Chapter 20 INTRODUCTION TO DIFFERENTIAL CALCULUS
Find the first derivative:
y = x^'X
In y = — In x
J x
1
"■ y
y
— -7— [In x] + In x — [x-1] =
x dx
dx
xl
y =
y
(1 -lnx) x^d-lnx)
xA
X1
In x
xl
y = x*nx
In y = In (xln x) = In x In x = (In x)2
1
"■ y
y
In x ln x /rt, x 1
+ = (2 lnx)-
•A/ «A/ «A/
y =
2 (lnx)xln*
x
= 2(lnx)xln^-1.
y
•A/ •
= V* ln x
lny
- y = V*
a
y
— + In x
A
V
x
V
±r-l/2
2
1 ln x
+
2 + lnx
V* 2^[x 2^x
y
= x^
(2 + ln x
V
2>/x"
y
= (^)*:
In 3/ = (x2) ln \x
~~ y
y
(x2) -7— [in V*J + 0n V*/ 2x
(*2)
Vx
— ]x"1/2 + 2 x ln V* = — + 2 x ln V*
V
x
— + x ln x ;
y = y
— + x In x 1 = VvxJ
v^ )
2 A
— + x In x
V2

Section 20.4 Special Techniques of Differentiation
707
y
lny
_ x cosh x
r =
In \x cosh x ) = cosh jc5 (In x) .
cosh*5 ._ w . . ^ ^ a
- + (In x) (sinh x°) • 5jc4
y = y
X
cosh jc5
X
+ 5x4 (In x) (sinh jc5)
_ (x cosh x J
cosh JC5
JC
+ 5x4 (In x) (sinh jc5)
Differentiation of functions involving several factors is often
more easily handled by considering the natural logarithm of
the function.
_r2 . /1 - x2
determine y1 by logarithmic differentiation.
Take the natural logarithm of both sides,
.^2 fiTI
V l+x
2
2"
Iny = In I e"
= -x2 + | {in (1 - x2) - In (1 + x2)}
Differentiate both sides with respect to x,
)dx- ^X + [2j1_x2 [2)1+x2
Solve for dy/dx,
2x(x4- 2)
(1-x4)
dy _ 2 x (x4 - 2)
d* " y (1 -jc4)
and replace y by e
-*2 ^/1 ~*2
\1+*2 '
dy _y2^ /l -*~
dx \ 1 + jc2
2 x (x4 - 2)1
I (1-x4) J
JC
g (x) = [ 1 + — J ; determine gy (x).
In g(x) = ex\n\ 1 + —
1 d
— gy (x) = ex
dx
In I 1 + -
x /J
gy (x) = ex
In 1 + -
X J x (1 + x)
+ ln[ 1 + — I e*
x
x

708 Chapter 20 INTRODUCTION TO DIFFERENTIAL CALCULUS
Implicit Differentiation
If in f (x, y) - 0 - a function with variables x and y — the
variable y is thought of as dependent on x ("in principle but not
in practice, a function within /*"), then f(x,y) = 0 is said to
define y as an implicit function of x.
The process of determining the derivative of one of two
variables with respect to the other is known as implicit
differentiation. In the above f(x,y) - 0, the derivative dy / dx is
determined by treating y as an implied function of x as the entire
function is differentiated.
To determine dy/dx by implicit differentiation, take the
derivative of each term of the equation and remember that y
depends on x. Terms involving both variables call for the use
of the product rule.
Implicit differentiation of a function given by the equation
x2+y2 = 1,
where y is thought of as depending on x, thus implying the use
of the chain rule when differentiating y2, gives
2x+2yf^.y0
dy\ x_
dx) " y
In this case, it is also possible to differentiate the explicit forms
of the equation, and thus pursue the solution by ordinary
methods of differentiation of the explicit functions
y = yl -x2 ; y = -yl -x2 .
If, however, the equation of an implicit function cannot be
rendered into explicit form, implicit differentiation is
indispensable.
x2 + x2y + Sxy + y2 = 0 ; determine -:— .
Differentiate with respect to x:
[x2 + x2y + 3 x y + y2] = 0
2x +
dx
x2 I TT ) + y ' 2 x
> + i
3,[^| + 3y
dx
Solving for dy /dx gives
dy 2 x + 3y + 2 xy
&x ~ x2+Sx + 2y
♦*,(&!-o

Section 20.4 Special Techniques of Differentiation
709
x2 _ yl _ 4 . determine
dy , d2y
-7— and r
dx dx2
by implicit differentiation.
2x-2y -^-= 0
dx
i
y
X
y
d2y _ dy' = j^xy^
dx2 dx
r
y-x
y
y2 - x2
dy d2y
xy = ey ; determine -r— and —r- .
d* dxz
The equation cannot be explicitly solved for y, and we use
implicit logarithmic differentiation.
In (x y) = In ey ; In x + In y = y.
d n i d n dy
jc y ydx
dy
dx
y =
d^y
dx2
y
x (y - 1) '
= -^ = -1
dx dx
y
x (y - 1)
In y' = In y - [In x + In (y - 1)]
= In y - In x - In (y - 1) .
Differentiating with respect to x :
7d*2 y ^; "
dx2
£y
dx2
(y')
X
M2
y(y - i)
(y-D
y_
X
iy')
Insert y
thus,
x (y -
d2y
dx2
1) '
2y+y3-2y2
x2 (y - 1)3
y(2+y2-2y)
x2(y- 1)3

710 Chapter 20 INTRODUCTION TO DIFFERENTIAL CALCULUS
20.5 Partial Differentiation
Differentiation with Respect to One Variable
In partial differentiation a function of more than one
independent variable, y = f(x, y, z, t, ...), is differentiated with
respect to one of these variables, the other variables being held
constant. The result of the process is a partial derivative.
The partial derivative of the function f (of several variables
x,y, ...) with respect to x is denoted
t-- or dfl dx .
The partial /i-th derivative of the function f (of several
variables x,y, ...) with respect to x is denoted
dnf
r-j- or dnfl dxn .
dx11
The symbol d ("mirror 6") denotes partial and seems to have
first been proposed in 1788 by the French mathematician
Lagrange. It must not be confused with the Greek letter 5
(lowercase delta).
For clarity, a subscript may be used to indicate the variable or
variables that are held constant, e.g.,
Consider the function
w = f(x,z) = Jt2 + JtZ2.
The partial derivative of w with respect to x is obtained when z
is held constant; thus,
$HL - o 2
"^—™" — <Ci X t Z
dx
In the same way, the partial derivative of w with respect to z is
obtained when x is held constant; thus,
dw
-^ — <Ci X Z .
dz
The principle of partial differentiation is readily understood
if we study the influence of individual variables on any
functional relation of several independent variables.
Volume V of a right cylinder varies with its height h and
diameter d; thus, volume is a function of height and diameter.
To find how the function
V(h, d) = n
(dYh
[2)h
is affected by changes of the independent variables, we
consider these variables one at a time.

Section 20.5 Partial Differentiation
711
The rate of change of V with respect to d, when h is held
constant (the partial derivative of V with respect to d) is
dV\ nh nh
dd)h = — -2-d=—-d'
and the rate of change of V with respect to h, when d is held
constant (the partial derivative of V with respect to h) is
fdV\ _ nd2 _ nd2
d
Partial differentiation follows the basic rules of ordinary
differentiation, but it must be kept in mind which variables
are held constant.
f(x, y) = x3 + 3 x2y - 2 x y5 ; find the first and second partial
derivatives of/"with respect to x andy.
df d2f
To determine -r- and T"2T, ^e vai*iable y must be held constant.
j£- [xs + 3x2y-2xy5] = 3x2 + Sy (2)x - 2y5(l)
= Sx2 + 6xy-2y5 .
d2f d
n - -, [Sx2 + 6xy-2y5] = 6x + 6y(l)-0
dx2 dx
= 6x+6y.
df d2f
To find t— and —r-, x must be held constant.
dy dy2
— [x3 + 3x2y-2xy5] = 0 + 3x2(l) -2x • 5y4
= Sx2 -10xy4.
d2f d
dy2 dy
[3x2 -10xy4] = 0-10x-4y3 = -40xy3 .
g (x, y) = sin 3 x cos 2 y ; find the second partial derivative of g
with respect to x, and with respect toy.
dg
t— = cos 2 y (3 cos 3 x) = 3 cos 3 x cos 2 y ;
d2g
—r- = 3 cos 2 v (- 3 sin 3 x) = - 9 sin 3 x cos 2 y .
dx2
dg
t— = sin 3 x (- 2 sin 2 y) = - 2 sin 3 * sin 2 y ;
—— = - 2 sin 3 x(2 cos 2 y) = - 4 sin 3 x cos 2 y.
dy2

712
Chapter 20 INTRODUCTION TO DIFFERENTIAL CALCULUS
Determine the first and second partial derivatives of
x
f(x,y)=xy; g(x,y) = —;and h(x,y)=xy
with respect to x and to y.
= 0;
= 0;
X
p. 701 T" =
K
dx
dg_
dx
3*
dy
dh_
dx
dh_
-v
dy
d2f
= y' d^
= I. d2s
y ' dx2
o
= -xy~z = ■
». i
= yxy-1;
= xy Inx ;
y
2 »
dy dy
= 0.
d2g
dy?
d2h
dx2
dh
dy?
= -x(-2)y
-3 _
2x
y3
= y(y-l)xy-2
= x^ln2 x .
Mixed Partial Derivatives
Second- or higher-order partial derivatives with respect to
several independent variables are called mixed partial
derivatives.
The notation
d2f
dy dx
implies that the function f(x,y) is first differentiated with
respect to x and then with respect to y; the notation
d2f
dx dy
implies that it is first differentiated with respect to y and then
with respect to x.
These notations may seem to be written backwards with
regard to the order in which the derivatives are determined.
To understand the notation, w& study the identities
d
dy
dx
The functioi
which g
ives
d2f
dy dx
i
w =
d_
dx
_dy_
d2f
dx dy
f(x,y) = x2y3,
rr— = 2xy3 and -^- = 3x2y2
dx J dy J
and the mixed partial derivatives
= r-[2xy3] = 2jc3y2 = 6xy2
and
dy dx dy
-^- = j- [3*2y2] = y2 • 3 • 2*1 = 6xy2
dx dy "x

Section 20.5 Partial Differentiation
713
The equality
d2w d2w
dy dx dx dy
is no coincidence; it holds good for every function w = f'(x, y)
d2w d2w
whose partial derivatives and are continuous
dy dx dx dy
functions.
The rationale behind the use of mixed partial derivatives is
illustrated by this example.
In an experiment, the volume V of a gas is a function of
temperature T and pressure p, according to the equation
cT
V{T,p) =
P
where c is a constant.
Sought I: The change in volume with respect to the change in
temperature.
dV__ d_
dT " dT
c T
P
c(l) c
= — ; p = constant.
P P
Sought II: The change in volume with respect to the change in
pressure.
r— = — [cTp-1] = cT(-l)p-2 = --T- ; T = constant.
dp dp ^ p2
Sought III: The change in volume with respect to changes in
temperature and pressure.
The same answer is obtained whether we determine
d2V d2V
or
dp dT dT dp
d2V d
dp dT dP
J)\ dp ' p2
g(x, y) = sin 3 x cos 2 y ; find
dx dy
First differentiate the function with respect to y, regarding x as
a constant,
sin 3 x (- sin 2 y) 2 = - 2 sin 3 x sin 2 y ;
then with respect to x, regarding y as a constant,
-2-3 cos 3 x • sin 2 y = - 6 cos 3 x sin 2 y .
Hence,
3¾
dx dy
= -6 cos 3 x sin 2 y

714 Chapter 20 INTRODUCTION TO DIFFERENTIAL CALCULUS
z=f(x, y) = e*^+ jc cosy
d2z
Sought:
dy dx
dz d2z d r n
_ = yexy+ Cosy; = 5- \y e ^ + cos y\ .
ox ^y 3X cry
~[ye^] = ye^^ [jc- 1+y-Ol+e^^. 1 = ay e^ + e**
d2Z
- xy e*y + e*y ■
dy dx
h =f(x,y) = e*tany
0 ^ *2h
Sought: a:
B dx2
a:
y = constant.
- sin y .
b: dy2
e x is its own derivative.
a2/i
c:
3y dx
^ d2h
d:
dx dy
dh
— = e* O + e* tany = e*tany.
—-- = e* -O + e* tany = e*tany.
dx1
b:
x = constant.
t— [e*tany] = ex sec2y + 0 • tany = exsec2y
— = e*2secy (^-[secy] J
= 2 exsec2y tany .
c:
= 3— [e*tany] = exsec2y
dy dx ^
d:
d2h
dx dy
- ex sec2 y .

Section 20.5 Partial Differentiation
715
w = f(x> y) =
X
arctan — ; find
y
d2w
dx dy
dw
dy
1 +
fx\* W
-11
y
y*
2 + x2 xdy
\yj
y
y2 + x2
x(-l)y-2 =
x
Thus,
d2W d
dx
Or l^-1])
x2 + y2
x
L x2+y2j
dx dy
and, by the quotient rule,
d2w (x2 + y2) - x • 2 x
dx dy
(x2 + y2)2
x — y
(x2 + y2)2 '
)
4b^j&*)£~'f^tg*

716
Chapter 20 INTRODUCTION TO DIFFERENTIAL CALCULUS
20.6 Mean-Value Theorems
fix)
A
Rolle's Theorem
Suppose f is a continuous function that crosses the x-axis at two
points a and b and is differentiable at all points between a and b
- that is, it has a tangent at all points on the curve between a
and b. Then there is at least one point between a and 6 where the
derivative is 0, and the tangent is parallel to the x-axis.
The function f shown below has four such points ci ... C4between
a and 6:
fix)
fix) = 0
fW = 0
f'ix) = 0
x
X
ROLLE Michel
(1652-1719)
f'ix) = 0
The theorem is known as Rolle's theorem after the French
mathematician Rolle. In more stringent mathematical
language, the theorem can be stated:
If a function fix) is continuous in the closed interval [a, 6]
and differentiable in the open interval ]a, 6[, and if
f(a) =fib), then there exists in the interval at least one
value c such that f ' (c) = 0 (a < c < b).
LAGRANGE Joseph Louis
(1736-1813)
Lagrange's Mean-Value Theorem
Rolle's theorem is a, special case of the mean-value theorem,
also called Lagrange's mean-value theorem after the Italian-
born French mathematician Lagrange, who was the first to
state it:
A function fix) that is continuous in the closed interval
[a, 6] and differentiable in the open interval ]a, b[ has in
this interval at least one value c such that
f'ic) =
fib)-fja)
b - a

Section 20.6 Mean-Value Theorems
717
Restricted Mean-Value
Theorem
Proof:
Specifically this means that the slope of a secant which
transects the curve of a continuous function at two points a and b
is the same as that of a tangent to the curve at some point c
between a and b.
To distinguish the Lagrange theorem from the more general
Cauchy mean-value theorem, which is concerned with two
intersecting curves, the Lagrange theorem is sometimes called
the restricted mean-value theorem, as one of the curves must be
a straight line.
The slope (m) of the secant line g(x) transecting the curve of the
differentiable function f(x) at the points {a, f(a)} and {6, f(b)} is
f(b) -f(a)
m —
We have, for all x,
m =
b - a
g(x)-f(a)
x — a
which gives the equation of the secant,
,,, f(b)-f(a) ,
g(x) = f(a) + r (x - a) .
b — a
We construct a support function h (x\
h{x) t= f(x)-g (x)
f(b)-f(a)
b - a
which intersects the x-axis at the points a and 6,
h (a) = h (6) = 0 .
As h (x) is the difference between the differentiable functions
f(x) andg(x), it is also differentiable,
f(b) -f(a)
= fix)
(x-a)-f(a),
h'(x) = f'(x)
a
Letting
h (c) = f (c)- 7 = 0
a
we see that there exists a point c such that
fib) -f{a)
/'(c) =
b - a
a < c < 6 .

718
Chapter 20 INTRODUCTION TO DIFFERENTIAL CALCULUS
A secant to the curve
f(x) = -x2 + Sx + 15
is defined by the points (-2, -5) and (7, 22); find the point where
the tangent of fis parallel to the secant.
The slope of the secant is
22 - (- 5) 27
msec -7_(_2)-9_d-
The derivative of f (x) is f ' (x) = - 2 x + 8.
The condition
3 = -2jc + 8
gives
x = 2.5 ; f(x) = 28.75 .
fix)
A (2.5,28.75)
30 -
(-2, -5)
x
f(x) =-x2 + Sx + 15
Cauchy's Mean-Value Theorem
The generalized mean-value theorem applies to any two
continuous and differentiable functions intersecting at two
points. It can be stated thus:
If two functions f and g, which have the same value at x = a
and x = 6, are continuous in the closed interval [a, b] and
differentiable in the open interval ]a, b[, and if g (b) -g (a)
* 0 and g' (x) ^ 0 in ] a, b[, then there exists in ] a, 6[ at least
one number c such that
f(b)-f(a) _ f'(c)
g(b)-g(a) ~ g'(c)
a <c <b.
CAUCHY Augustin Louis
(1789-1857)
The generalized mean-value theorem is also called the
extended mean-value theorem, or Cauchy's mean-value theorem,
after the French mathematician Cauchy.

Section 20.6 Mean-Value Theorems
719
Proof:
g(x)
X
a <c < b
By observing that
f(b)-f(a)
= 1 =
g(x) - f(a)
g(b)-g(a)
we find the equation
f(b)-f(a)
g(x) - g(a)
g{x) = f{a) +
{g(x)-g(a)}
g(b)-g(a)
As before, we construct a supporting function h (x),
h(x) = f{x) -g{x)
= f(x) - f(a)
f(b)-f(a)
g(b)-gja) «*<*>-*<«»■
h(a) = h (b) = 0, and h (x) is differentiate on ]a, b[
h'ix) = fix)- f(b)-f(a) S<(X)
Letting
h'ic) = 0,
we find a point c in ] a, b[ where
/(6)-Ao) _ f'(c)
g(b)-g(a) ~ g'(c) ' a<c<6'
LECONS
SCR LE
CALCUL DIFFERENTIEL,
PAR M. AUGUSTIN-LOUIS CAUCHY,
maiNiiiaii rj crr* tu »o.it» Nr CRAimiR*. tRotRURUR al'Icolii rotalr roLiTiCRdiqaR.
VMtRURBR AMOIRT A LA fACBLT* 01* ICIRJICR* , RRMRRR »R l'aCAAIUIR DM lCISRCaj,
CBRTALIRR RR LA UttlOJ B*ROS>RDB.
A PARIS,
CBEl OB >UBI FH±RBS, LIBRAIRES 00 ROI ET Of LA B1BLIOTBZQUE 0U ROI,
act tfRrtfiTR, a.* 7.
1829.
Cauchy gave elementary calculus the character it still bears today.

721
Chapter
21
INTRODUCTION
TO INTEGRAL CALCULUS
21.1
21.2
21.3
21.4
21.5
*** 32.4
Basic Concepts
Methods of Integration
The Definite Integral
Multiple Integrals
Improper or Unrestricted Integrals
Numerical Integration
Page
722
723
747
753
756
947
Indicates cross-reference

722
Chapter 21 INTRODUCTION TO INTEGRAL CALCULUS
21.1 Basic Concepts
Integrand
Integral
Latin, integer, "complete";
integralis, "making up a whole"
von LEIBNIZ Gottfried Wilhelm
(1646-1716)
Arbitrary Constant
Integration Constant
Indefinite Integral
Antiderivative
Primitive Integral
Definite Integral
In Section 21.3 (p. 747) we
will define the definite
integral and see that
b
J/(c) dx = F(b)-F(a)
a
for any integral F off.
In differentiation, relatively simple and straightforward
rules enable us to find the derivatives of even quite
complicated functions. In integration, which is the reverse process of
differentiation, our task is to find a function whose derivative
is known. This often involves intelligent guesswork, and it
might still not be possible to integrate a given function. When
we have arrived at an integration formula, it is always
possible, however, to verify it by differentiation.
The function to be integrated is referred to as the integrand;
the result of an integration is an integral.
The integral sign J, an elongated S denoting sum (Latin:
summa), was introduced by Leibniz, who named integral
calculus calculus summatorius.
If we differentiate the functions
f ix) = x3 + x ; g(x) = x3 + x + 4 ; h(x) = x3 + x - 7
with respect to the independent variable xt we obtain the results
fl(pc) = g'(x) = hl ix) = 3 x2 + l
in all three cases.
If we reverse the process and integrate the expression Sx2 + 1,
we obtain jc3 + x, but the constant terms of the functions g and h
cannot be recovered failing information as to initial or
boundary conditions. Therefore, the integral must be written
J (3 x2 + 1) dx = Xs + x + 4 + C,
where C is an arbitrary constant or integration constant.
The notation dx indicates that the integration is to be
performed with respect to the variable x\ adu would indicate
integration with respect to a variable u,etc.
A function F is an integral of a function f(x) if F' ix) = f(x) for
all x in the domain of f, and, conversely, for every value of a
constant C, F(x) + C is also an integral of fix).
An integral with no restrictions imposed on its independent
variable is known as an indefinite integral, antiderivative,
or primitive integral,
jf(x) dx,
whereas an integral that is defined by the limit values a and b
of the independent variable is a definite integral,
b
\ fix) dx .
a
Calculating the integral of (3 x2 + 1) from 2 to 3, we find
J(3 x2 + 1) dx =
ix3 + x + C)
= (27 + 3 + 0-(8 + 2 + 0 = 30-10 = 20,
from which we see that a definite integral is independent of the
arbitrary constant in the corresponding indefinite integral.

723
21.2 Methods of Integration
Integrating Power Functions
Differentiating xn with respect to x gives nxn ~1; analogously,
xn+1 yields (n + 1) xn, and consequently jxn+l becomes xn.
Thus, if -7— = xn, we havey = =-xn+1 + C, that is, the integral
n+1
x
of xn with respect to x is =-, or
^ n + 1 '
x
n+1
\xn dx = r+C,
J n + 1 '
where n^-1 and C is a constant.
p. 699
Integrate:
. f*3/2d* = f*5/2 + C
J 5
2
• j ^fx dx = jx^2 d x = -^x3/2 + C
C 1
dx = Jjc-^d jc = 2^x + C
3 Q
J V^c dx = jx1/3 d x = — xm + C
x
n +1
For x~~1, the power rule \xn dx = + C leads to the expres-
*.0 i
sion — = — . A search through various derivatives reveals that
d 1
1— [Inx] = —. Hence, we have
ax x
J jc"1 dx =lnx + C;jc>0.
More generally, since -=— [In | x \ ] = x~ \
jx"1 dx = In |jc| + C; x^O .
Fundamental Arithmetic Integration Rules
A constant factor in the integrand
sin x - 5 cos 2 x dx = 5 sin x • cos 2 x dx
or, generally,
j k f (x) dx = /j J/*(jc) dx, where k is a constant,
can be taken out and placed before the integral sign.

724
Chapter 21 INTRODUCTION TO INTEGRAL CALCULUS
When differentiating a sum, we differentiate each term
separately; reversing the process, we integrate each term
separately,
f \f (x) ± g (x)] dx = jf(x) dx ± jg (x) dx
or
J (3 x - 5) dx = S\x dx - 5 J dx
= s(y+C1)-5(x + C2) = |x2-5x + 3Ci-5C2,
where (3 C\ - 5 C2) is simply another constant:
3
J (3 x - 5) dx = -rx2 - 5 x + C .
Basic Integration Formulas
Every differential formula, written in reverse, gives an
integral:
■j— [sin jc] = cos x ; Jcos x dx = sin x + C .
We can conveniently reverse differentiation formulas to
establish a list of basic integration formulas. These, or
modifications, form a starting point in the various techniques
of integration. Below, t is a variable, a and n are constants.
tn+\
tndt = 7+C; n*-l.
n + 1 '
r1 d* =
/
Jdt-
J J
/ d W ^
d*
V * J
dt = In |£| +C; £*0.
e'd£ = e' + C and \eat dt = - + C.
3 a
a1 dt =
a1
In a
+ C; 0<a*l.
sin at dt = -— cos at + C.
a
cos a£ dt = — sin at + C .
a
sec at dt = — In I sec at + tan at\ + C
a ' '
sec2 a£ d£ =
esc2 at dt =
r 1 1
- dt = f(l + tan2 a t) dt =— tanat + C
cos2 at a
d t = - — cot at + C.
sin2 at a
r 1 t
dt = arcsin —+ C; a > 0 .
V^T7
a

Section 21.2 Methods of Integration
725
r i l t
— - dt = — arctan —+C.
a2 + t2 & &
\sinh at dt = — coshat + C.
J a
f cosh at dt = — sinh at + C .
J a
W±a^
r 1
dt = \n |* + V*2±a2 | +C
r 1
V*271
d£ = sinh-^ + C.
d£ = cosh-^ + C; \t\ >1.
— d£ =- tanh"1-+C = t—In
a2 -t2 a a 2 a
a + t
a
+ C.
p. 722
The above formulas can all be checked by differentiation.
Tables of Integrals
Tables of some thousand basic integration formulas have been
collected over the years. A comprehensive table is often of
great help, but no matter how extensive, it will only
sporadically give us exactly the needed formula.
Non-Integrability
The existence of a certain elementary function does not
necessarily imply that there must exist another elementary function
of which the given function is the derivative.
Some functions, such as
sin x
x
COS X
X
dx J sin(x2) dx
dx
I yx sin x
dx
Je~*:
dx
f—
J x
dx
x
dx ,
do not possess integrals that can be expressed by a finite
number of elementary functions.
Certain radical polynomial expressions (e.g., 1 - xl dx)
can be expressed by a finite number of elementary functions,
while others (e.g., J V 1 -*3 dx) cannot.

726
Chapter 21 INTRODUCTION TO INTEGRAL CALCULUS
Simple Rearrangements
p. 725
It is often possible to integrate functions directly by using
existing tables of integrals. The more comprehensive these
tables, the more likely one is to be able to integrate the function
directly. Quite simple rearrangements within the integrand
will often make it better adapted for integration.
,3/4
r 1 r*374 4
r— dx = \x~1/4 dx = ^-+C = £*3/4 + C
3
4
V*2 - 4
dx = 7
; dx = 7 In \x + V*2 - 4 + C
V*2 - 4
p. 724
Adding and Subtracting the Same Quantity
We may add a quantity at one end of the integrand and
subtract the same quantity at the other end; at first glance this
might seem rather bizarre, but it can serve a purpose.
Integrate J tan2 x dx .
| tan2 x dx = {(1 + tan2 x - 1) dx = {(1 + tan2 x) dx - j dx
= | sec2 x dx - | dx = tan x - x + C .
Breaking Up Fractions into Parts
A fraction may often be written as a sum of simpler fractions
that can be integrated directly.
f*3x3 + 2x2-6 + 9x sin x
3x
dx
= | jc2 dx + — |jtcbc-2J x"1 dx + 31 sin jc dx
= —jc3 + — x2 - 2 In | x | - 3 cos jc + C .
2 x - 3 J J 2
Completing the Square
Several basic integration formulas involve the sum or
difference of two squares. By completing the square, we can
often extend such formulas to integrands that contain a
second-degree polynomial.

Section 21.2 Methods of Integration
727
Completing the square: p. 310
The polynomial (ax2 + bx + c ) is "completed" in the following
manner:
o Factor the coefficient a from the rest of the expression:
ax2 + bx + c = a\ x2 + — x + —
a a
o Add and subtract the square of half the coefficient —:
a
ax2 + bx + c = a
2 b
x* + —x +
a
b\2
2a
Pi'*'-,
2 a J aj
o Regroup the terms
ax2 + bx + c = a'
2 b (b^2
x* + —x + ——
a \2a
c fbY
Depending on positive or negative sign of the radicand,
write, respectively
2
2 u I b Y k/ b2 "2
axz + OX + C = a\ x + -— J + I A/ c - —
or
ax2 + bx + c = a(x+ ^j - [^j - c + ^ )
p. 724
To integrate
dx ,
x2 - 6 x + 13
complete the square of the denominator of the integrand,
jc2-6jc+13 = (jc2-6jc + 32-32+13) = [(x - 3)2 + 4] .
So,
x2-6x+ 13
1
dx -
(x - 3)2 + 4
dx
(*-s)»;**"5arctaniTi+c-
p. 513
Use of Trigonometric Identities
Trigonometric identities that transform products into sums or
differences are often useful to adapt an integrand for direct
integration.
J sin x cos 2 x dx = I -r [sin (x + 2 x) + sin (x - 2 x)] dx
= -r J (sin 3 x - sin x) dx
1 !
= -r cos x - 77 cos 3 x + C .
p. 511
Jcos2jc dx = 7j J(l+cos 2 x) dx = Trx + jsin 2x + C

728 Chapter 21 INTRODUCTION TO INTEGRAL CALCULUS
Integration by Parts
Integration by parts is the reverse of differentiating a product,
where
\f(x)g(x)]' = f(x)g'(x) + g(x)f\x)
leads to
f{x)g{x) = $ f'(x)g(x) dx + \f(x)g'{x) dxs,
which may be rearranged
j f(x)gy (x) dx = f(x)g (x) -\g (x) f (x) dx.
By introducing u (x) = f(x) and v (x) = g (x), this formula may
be rewritten
juv'dx = uv—jvu* dx,
where we choose u and v so as to make the new integral easier
to integrate than the original.
We note the special case
judx = ux - \x ux dx .
The decision as to which part of the integrand to choose for u
and for v' is often a matter of trial and error. However, if one
part of the integrand is more complex than the other, it is often
good practice to choose that part as v'.
Integrate J x cos x dx.
We decide on
u = x
V ' = COS X
uy = 1
v = sin x + C\
\ x cos x dx = x (sin x + Cj) - J (sin x + Ci) (1) dx
= x sin x + C\ x + cos x - C\ x + C
= x sin x + cos x + C . •
The constant C\ will cancel out in the final result; therefore,
when integrating by parts, we may drop the first integration
constant.
Integrate J x In x dx.
Let u = In x and v' = x: then u' = — and v = -r x2 .
X 2
1 1 r
J x In x dx = rx2lnx-7
2
j
x2 — dx
x
= -T x2lnx--~ \ x dx
1 x2
= — x2 In x - — + C
2 4

Section 21.2 Methods of Integration
729
An integrand composed of only one factor may be considered
to include a factor 1 to permit integration by parts.
Integrate J In x dx.
Let u = In x and v' = 1; then u' = — and v = x .
x
f In x dr = jclnx- x — dx = xlnx - f dx = x (Inx - 1) + C.
J x J
The integration of a rather ordinary-appearing expression
will sometimes offer unexpected difficulties and surprises.
Integrate ^1 - x2 dx.
Let u = Vl -x2 and v' = 1; then w' = - . and v = x .
Vl - x'
j V1 -x2 dx = x^l-x2 +
r x2
dx .
Rewrite the numerator as a difference [1 - (1 - x2)] and expand
the integral into a difference of two integrals; thus,
j V1 -x2 dx - x^ll-x2 +
1-(1 -x2)
Vl - x'
dx
= x^l-x2 +
ii^x2
dx -
1-x2
Jl^2
dx
or
= x Vl - x2 + arcsin x - V 1 - x2 dx + C\
The original integral is now on the right of the equality sign
and it might be thought that we have reached a dead end. To the
contrary, move it to the left side of the equality sign:
2 Vl -x2 dx = x Vl -x2 + arcsin x + C\ ;
V 1 -x2 dx = — xyl-x2 + — arcsinx +C

730 Chapter 21 INTRODUCTION TO INTEGRAL CALCULUS
Successive Integration by Parts
Successive integration by parts is sometimes required to
complete the integration.
Integrate J e* cos x dx.
Let u - e* and v' = cos x; then u' = e* and v - sin x .
Integrate by parts:
(1) \ ex cos x dx = e* sin x-\ ex sin x dx + C\
To integrate by parts, let U\ - eP° and vi = sin x; then u\ = e*
and i>i = -cos jc:
J e* sin x dx = - e* cos x - j ex (- cos x) dx + C2
(2) = - e* cos x + j ex cos jc dx + C2
Adding (1) in (2) and rearranging terms, we find
J e* cos x dx = — e* (sin x + cos jc) + C .
Reduction Formulas
A reduction formula (or recursion formula) in integration
expresses a given integral as the sum of a function and a
known integral.
Several formulas in integration tables normally appear in the
form of reduction formulas, some of which are given below.
• Show that
J xn ex dx = xn e* - n \ xn ~ 1 e* dx + C ,
where n is an integer.
Let u = xn and v' = e*; then u' = n xn ~ 1 and v = e* .
Integrate by parts:
j xn ex dx = xn e* - n J xn ~ 1 e* dx+ C
• Integrate J x2 e* dx.
Using the reduction formula, we obtain successively:
J x2 e* dx = x2 eP0 - 2 j x2 ~ 1 ex dx + Ci
= x2ex-2 \x ex - J e* dx J + Cx + C2
= jc2 e* - 2 [jc e* - e*] + Cx + C2 + C3
= e*(*2-2:x; + 2) + C

Section 21.2 Methods of Integration
731
Show that
{ xn sin x dx = -xn cos x + n J xn ~ 1 cos x dx + C,
where n is an integer.
Let u = xn and v' = sin x; then u' = /i xn ~ 1 and i> = -cos x.
Integrate by parts:
{ x71 sin x dx = —xn cos x - \ n xn ~ 1 (-cos x) dx+C
= -x'1 cos x + n | xn ~ 1 cos x dx + C.
Show that
| x71 cos x dx = xn sin x - n $ xn ~~ * sin x dx + C,
where /i is an integer.
Let u = xn and v' = cos x; then w' = n xn ~ 1 and v - sin x.
Integrate by parts:
| x71 cos x dx = xn sin x - n j xn ~ 1 sin x dx + C .
Show that
{ (In x)" dx = x (In x)" - n \ (lnx)" -1 dx + C,
where n is an integer.
{ (In x)n dx = f (In x)" • 1 dx.
Let u = (In x)n and i/' = 1; then i/' = n (In x)
Al - 1
{xj
and
i/ = x.
Integrate by parts:
{ (In x)n dx = x (In x)n -
x n (In x)
n - 1
(1
^X
dx + C
= x (In x)" - n \ (In x)" " 1 dx + C.
Show that
| sin71 x dx = -
cos x sin71
/i
+ f sin71 " 2 x dx+C,
n J
where n is an integer, ^ 0 .
| sin71 x dx = | sin x sin71 ~ 1 x dx.
Let u = sin71 ~ 1 x and i>' = sin x; then
u' = (/1 - 1) sin71" 2 x cos x and i/ = - cos x.

732
Chapter 21 INTRODUCTION TO INTEGRAL CALCULUS
Pythagorean Identity:
p. 508
Integrate by parts:
cos x sin71 1 x
J sin x sin71 ~ 1 x dx -
— \ (n - 1) sin" " 2 x cos x (-cos x) dx + C
= - cos x sin" " 1 + (n - 1) J sin" ~ 2 jc cos2 x dx + C
= - cos x sin" " 1 + (n - 1) J sin" ~ 2 jc (1 - sin2 jc) djc + C
= - cos jc sin" " 1 + (n - 1) J sin" ~ 2 jc djc
- (n - 1) J sin" jc cbc + C.
Add (n-l)j sin" jc djc to both sides of the equation:
n \ sin" jc dx = - cos jc sin" ~ 1 + (n - 1) J sin" ~ 2 jc djc + C.
Dividing both sides by n gives
J sin" jc dx =
cos jc sin
n
n-l
n-1
n
\ sin" " 2 jc djc + C.
Analogously, we have
sin jc cos"
J cos" jc dx =
n
where n is an integer, ^ 0 .
— + f cos" " 2 jc cbc + C,
n J
r^o^A^
««T>
77/ y///V when it stops being fun,M
Drawing by M. Twohy © 1994
The New Yorker Magazine, Inc.

Section 21.2 Methods of Integration 733
Integration by Substitution
By replacing part of a composite integrand with a new
variable, the integrand may be rendered suitable for
integration by a basic integration formula. This technique is
the reverse of the chain rule for differentiating a composite
function.
If F is an integral of/*, then the chain rule gives
57 [F[g{x)\\ = f {g(x)}g>(x)
or
I fig U)) g'(x) dx = F{g(x)} + C .
o To apply the method of integration by substitution, select
within the integrand a function of x, say, g(x), to be
replaced with a new variable, say, u.
This selection should be made so that the new integrand
in u can be integrated on sight.
o Differentiate u with respect to cbc, and solve for dx ,
— - -n H du
dx ~ 8 W' ^ " g\x) '
du
o Make the substitutions g (x) = u and dx =
g\x) •
o Place constant factors before the integral sign.
o Eliminate all variables x. If this is not possible, try
another substitution or another method of integration.
o Integrate.
o Replace u by g (x).
Integrate x2 V*3 - 2 dx.
q ^ i du rt 9 _ _ dw
Let w = jr3 - 2; then 3— = 3 xL and cbc =
dx 3x2
We now have
du r
I jc2 V*3 - 2 cbc = x2 Vw
3x2
0 1/2 dt/
3x2
= jV/2f-= \\u^du = I.|.MM + C.
To obtain the answer in terms of*, replace u by (a;3 - 2),
J*2V*3-2 dx = |(x3-2)3/2 + C.

734
Chapter 21 INTRODUCTION TO INTEGRAL CALCULUS
Integrate [jce-* dr.
p. 724
Let u = —x2; dx =
du
2x
f u du 1 f u ^
\ -x e " = - — \ x e" —
J 2x 2 J x
- Je" dw = --e" +C
Going back to x gives
xe ^
dr = - — e-* + C.
Integrate J (jc5 + jc)10 (5jc4 + 1) dx.
Let w = jc5 + x ; dr =
dw
du
5jc4 + 1
w10 (5jc4 + 1) — = \u10 du
5 x4 + 1 J
Going back to x gives
10+1+C " 11 +c
J (jc5 + jc)10 (5x4 + 1) dc = —(x5+x)n + C
Integrate
(In jc)8 x
dx
Let u = In x ; -5— = — and dr = jc du
ax x
x du = J w~8 dw = —
-8 + 1
U8X
8+1
+ C =
-ju-UC
Going back to x gives
r 1 1
— dr = - — (In jc)~7 + C
(lnjc)sjc '
Integrate J sin jc cos jc djc.
Let u = sin jc ; dx =
du
cos JC
u cos a; = u du
cos x '
«2
T + c
2* sin2 jc + C

Section 21.2 Methods of Integration
735
Integrate x ^|x— 1 dx.
du
Let u = x - 1; then -:— = 1 and dx = du .
dx
j x V*-l dx = j(u + l) w1/2 dw = J w3/2 + um du
1/5/2 „3/2
+ c.
Going back to x:
x ^|x
1 dr= 2
(V^T)5 (V^Hl)
+ c.
When integrands contain different roots of the same quantity,
it is generally advantageous to use a substitute for the root that
is the highest common divisor of the roots involved, and
thereby eliminate all radicals.
Integrate
V* + V*
dx
Let u = V*; then u12 = x , yx = u4 , V* = w3, and dx = 12 u11 du
We now have
-\[x + -\[x
dx= 12
r u*
u + 1
du .
12
/• 1/8
u-1
du = 12
/
u1 + u6 +u5 +u4 +us +u2 + u + 1 +
V
u-1
du
To evaluate
u-1
du, let u - 1 = z; then du = dz, and
u-1
du =
- dz = In \z\ + Ci = In 1 - u I + C
Hence,
yx
dx = 12
fu8 U1 U6 U5 U4 US U2 ,1-
lT+T+6"+T+T+T+T+I/ + lnU-1
Going back to x gives
l""l| + C-
12
/-2/3 -7/12 rl/2 -5/12 rl/3 rl/4 rl/6 . 10 , A
lV+V+ir+V+W+V+*1/12+lnl ^-11 )+c

736 Chapter 21 INTRODUCTION TO INTEGRAL CALCULUS
Combining Substitution with Integration by Parts
Integration by substitution is often combined with integration
by parts.
• Integrate J arc sin x dx.
p. 724 Let u = arcsin x and v' = 1; then u' = , and v = x .
J arcsin dx = x arcsin x
x , dx
Vl-*2
= x arcsin x - J x (1 - Jt2)~~1/2 dx ; -1 < x < 1
To integrate J x (1 - jc2)~1/2 dr, substitute:
0 dw « , dw
m; = 1 - jcz; -5— = - 2 x ; dr = - -— .
dx 2 x
$ x (1 - *2)-i'2 d* = f x w~1/2 (- 5^) = -£J w-1/2 da;
2jc
= - w
1/2
+ Ci = -(I-jc^ + Ci
= -Vi- x2 + Ci.
Consequently,
J arcsin jc dr = x arcsin x + Vl - x2 + C.
Integrate e^x dx.
Let z = yx ; then -5— = — jc_1/2 and dr = 2 x1/2 dz .
dx 2
Since z = ^|x= x^2 ,
dx = 2z dz.
J e^ dx = 2 J z ez dz.
Let w = z and u' = ez ; then w' = 1 and v = ez.
2 \ ezz dz = 2L ez - J ez dz J = 2 (z ez - ez ) + C.
Returning to x gives
J e^ dr = 2 (V* e^ - e^) + C.
Integrate 7 J x2 e3 * dr.
Let w = jc2 and u' = e3 x ; then u' = 2 x .
To find f, integrate J e3x dr .
dz dz
Let z = 3 x: then -5— = 3 and dx = —
dx 3

Section 21.2 Methods of Integration
737
j e3x dx = ez — = — J e2 dz = o" e3* + C i.
3 3
WithCi = 0, v = |e3*.
Integrating by parts,
7jx2e3x dx = 7 [*2| e3*]- f | (e3*) (2 *) dx + C2
= -n x2 e3 * --o- \ x e3 * dx + C^
To repeat integration by parts,
let u\ - x and v\ = e3*; then u\ = 1 andi>i = Tre3*.
7 J jc2 e3 * dx = o"jc2 e3 * --^-
■(**■)-/*■>•*;
+ c3
7 14 14
^ bum /y^ 0*5 3C ^^^ /y pt*3 3C i ^^^_ |-vO 3C i /*
Successive Substitution
It is sometimes advantageous to use successive substitution.
Integrate
j Vi + V^
dx
j -i
Let u = ^[x ; then -r— = — x~^2 and dr = 2 jc1^2 dw = 2u du .
dx 2
JVl + V* djc = JV1 + u 2u du,
which still does not agree with any basic integration formula.
We make a new substitution,
z = 1 + u ; then du = dz.
The integral now becomes
z5/2 z3/2
JVl + w 2wdi/ = Jz1/2 2 (z - 1) dz = 4|
Replacing z with 1 + w gives
'(VTT^)5 (VTT^r
+ c
J V1 + u 2 u du = 4
V
7
+ C
and replacing u with V* results in

738
Chapter 21 INTRODUCTION TO INTEGRAL CALCULUS
Substituting Trigonometric Functions for Algebraic Expressions
Until now, our substitutions have all consisted of letting u
represent some function within the integrand. However, for
integrands that contain certain radicals, for instance, the
square roots ya2 - x2, ^x2 — a2, and ya2 + x2, a substitution by
a trigonometric function is sometimes more productive.
Consider the triangles
pp. 505 - 506
a
x = a • sec 0
Va2 - x2
We find that the substitutions
x = a • sin 0 x = a • tan 0
bring, by definition, the substitutions
V a2 - x2 = a • cos 0 ; ycfi + x2 = a • sec 0; yx2 -a2 = a • tan 0.
To ensure a one-to-one relation between the original variable
and the substitution, restrictions must be placed on 0. These
restrictions are the same as the range of the corresponding
inverse trigonometric functions:
Substitution
Range
x =
x —
X =
a sin 0
a tan 0
a sec 0
~2'2J
ya2-x2 = a cos 0 arcsin 0
ya2 +x2 = a sec 0 arctan 0 : I - —, — I
V*2 -a2 - a tan 0 arcsec 0 : I — 7C , — ^" I I 0, — I
Integrate
V4 - x2
dx
xl
C ^4^2
dx =
H22-x2
xA
dx.
xd
Let x = 2 sin 0; dx = 2 cos 0 dO
a/4 - x'
dx =
X'
C 2 cos 0
4 sin2 0
2 cos 0d0 =
f cos2 0
sin2 0
d0
= |cot20 d0 = |(csc20-l)d0 = -cot0-0+C.
We have
. x V4 - x2 V4 - x2
sin 0 = — ; cos 0 = ; cot 0 =
2 2 x
and obtain
f ^4^2
dr = -
V4 - jc'
JC
:r
JC
arcsin — + C.

Section 21.2 Methods of Integration
739
The example below shows how the integration of a seemingly
simple expression may require some ingenuity.
Integrate I V*2 + 3 dx.
p. 724
V^73"= ^x2 + (V3)2 .
Letx = V3tan0; dx = ^3 sec2 0 d0
V*2 + 3 = ^3 sec 0.
We now have
J V*2 + 3 dx = J (^3 sec o) • (^3 sec2 0) dO
or
= 3 J sec3 0 dO
3 J sec 0 sec2 0 d0,
which we integrate by parts; let u = sec 0 and v' = sec2 0; then
w' = sec 0 tan 0 and v = tan 0,
J sec3 0 d0 = tan 0sec 0- J sec 0 tan2 0 d0 + (¾
= tan0sec0-Jsec 0 (sec20- 1) dO + C1
= tan 0sec 0- J sec3 0 d0 + J sec 0 d0 + (¾
2 J sec3 0 d0 = tan0sec0+J sec 0 d0 +Q
r q ^ , ^ tan ^ sec 0 1 f ^,^ „
J sec3 0 d0 = 5 + - J sec 0 d0 + (¾
tan 0 sec 0 1,, ^ ^, „
5 + — In | sec 0 + tan 01 + C,
and
(^3 tan 0) ( ^3 sec 0) 3
3 J sec3 0 d0 = + - In I sec 0+ tan 0| + C.
/- V*2 + 3
Reintroduce V 3 tan 0 = x and sec 0 = —— to obtain
Vs
J
P2—- , x ^x2 + 3 3
V *z + 3 dx = +-ln
V x2 + 3 x
V3 V3
+ C
jc V*2 + 3 3 ,
2 +2ln
Va2 + 3 + jc
V3
+ C.
In the above calculations, we have stayed with the basic
integration formulas on pp. 724 - 25. Judicious use of a more
comprehensive table of integrals could simplify the process.

740
Chapter 21 INTRODUCTION TO INTEGRAL CALCULUS
Converting Rational Trigonometric Integrals into Algebraic Integrals
Every rational function of sin x and cos x can be converted
into a rational algebraic function.
pp. 518 -19
pp. 510 -11
To solve a trigonometric equation, we converted trigonometric
terms to the tangent of the half-angle; similarly, the substitu-
x
tion u = tan — to convert rational functions of sin x and cos x
into rational algebraic functions may prove profitable for
integration;
x x
u = tan- , sin - = —
With the identities
u
x
+ u
2'
and cos — =
VI
+ w
sin 0=2 sin — cos — ; cos 0 = cosz —- sinz —
x
and the substitution u = tan—, we obtain
sinx = 2
u
r
r
COSJC =
+ u
v2
V 1 + U2 W 1 + U2
f
WT
+ w
u
2u
1 + u2
l-«2
>
WT
+ w
1 + u2 '
and
du
dx
d_
dx
x
tan-
1(
2tSeC 2
2£Ll
l + tan2!j = i(l + w2)
dx =
1 + u2
du .
Integrate
sin x
dx.
x 2u 2
Let u = tan—; then sinx = — and dx = -
* 1 + u1 1 + u1
du
sin x
dx =
r
( 1 \
2u
\ 1 + U2 j
1 + u2
du
— du= In \u\ +C =ln
u ' '
tan |
+ C
Integrate
cos x
dx.
x
Let u = tan—; then cos x
1-u2
1 + u2
and dx =
1 + u2
du

Section 21.2 Methods of Integration
741
p. 725
COS X
dx =
f 1 \
1-u2
V 1 + u2 )
2 J ~C 1
du = 2
l + u2
1-k'
dw
= 2 I - I In
1 + u
1-u
+ C = In
1 + tan —
1 - tan —
+ C
Integrate
1 - sin x
sin jc — sin x cos jc
dx
1 - sin x
sin jc — sin x cos jc
x
dx =
1 - sin x
dx.
sin jc (1 - cos x)
2u 1-u2 2
Let u = tan—; then sin x = -, cos x , and dx = du.
* l + u1 l + u1 l + u1
1- sinx
sinx — sinx cosx
dx -
2u
1 + u2
2u
l-u2\ [l + u2
du
l + u2 V l + u2
= | \(u-1-2u-2 + u-*) du =| (In \u\ +2u-2-^u-1) + C
In
x
tan-
+ 2 cot — - - cot2— \+C
Integrating
fXx)
fix)
dx
Why'
An integral where the numerator is the derivative of the
denominator may be evaluated by the formula
r fix)
~, v dx - In I f (x) I + C .
fix) " ■
du
With f(x) =u,
and
f ix)dx = du, dx =
fix)'
/*/•■
fix)
rrx
dx =
fix) du
u fix)
dx =
— du
u
p. 723
— du
u
= In | u | + C (w * 0), and
/*/•■
fix)
dx = In |/*(jc)| +C
It is advantageous to watch for integrands whose numerator is
the derivative of the denominator, or some multiple thereof, as
they are easier to deal with.

742
Chapter 21 INTRODUCTION TO INTEGRAL CALCULUS
Integrate
2x
x2+l
dx
-7— [x2 + 1] = 2 X .
ax
Consequently,
r 2 x
— dx - In I x2 + 11 + C .
J x2+ 1 ' '
Since x2 + 1 is always positive, the absolute sign is not needed;
C 2x
1 +JC'
dx = In (x2 + 1) + C .
Integrate
f sec2 jc
tan x
dx .
dx
[tanx] =
sec2 x;
C sec2 jc
tan x
dx = In | tan x \ + C
Integrate { cot jc dr.
| cot jc dx =
cos JC
sin jc
dx = In |sinx\ + C .
Integrate
jc - 2
jc2 - 4 x + 3
djc
1
2
r 2 x — 4 1
— djc =;rln |jc2-4jc + 3| +C
jc2 - 4 jc + 3 2 I I
Integrating
dx
If the denominator of the integrand is the square root of a
function of the variable, and the numerator is the derivative of
the radicand, then
tML dx = 2^f(x) +C,
yffW
which we verify by letting f(x) = u:
_d_
djc
[2 ^U + C] = ^=r .
yu

Section 21.2 Methods of Integration
743
C 3 X2 + X ,
Integrate ==• ax
J V 4x3 + 2x2
Since
dx
[4x3 + 2x2] = 4 (3 x2 + x\
the integral has the form
f\x)
V7w
dx.
Hence,
' 3x2+x = l ^-g——^- =y2^j2x3+x2 + c
^4x* + 2x2 4 2
If the integrand does not immediately lend itself to evaluation
by the above formula, it may be rewritten in a manner such
that the formula can still be used.
Integrate
V¥
dx,
p. 724
V
6 — x
x
dx =
6 - x 1
, •=- dx = —
V6 x - x2 z
f(6-2x) + 6
1
2
f 6-2*
^16 x - x2
dx + 3
\6 x — x2
\6 jc - x2
dx
dx
= - • 2 • V6jc-jc2 + 3
= V6jc-jc2 + 3
V32-(x-3)2
• *~3 /-
arcsin —-— + C .
dx
To evaluate the integral
r a x + b
Vc x2 + d x + e
dx
by the formula
numerator thus,
dx = 2 ^j fix) + C, we rewrite the
<fix~)
p -j— [c jc2 + dx + e] + q ,
where p and g are constants.

744
Chapter 21 INTRODUCTION TO INTEGRAL CALCULUS
Integrate
C 4 x-2
V5 + 4x -x2
dx
p -7— [5 + 4 x — x2] + q
p (4-2x) + q
which gives
Hence,
4jc-2
4x-2,
P = -2; q
= 6
r 4*-2
\5 +4 jc - x2
dx =
f-2(4-2x) + 6
p. 724
V5 + 4 x - x2
f -2 (4-2 x)
V 5 + 4 x - x2
dx
dx +
= -2-2 V5 + 4^-j?+ Ci + 6
V5 + 4 x - x2
r 1
dx:
where
V32-(x-2)2
and, consequently,
f 4 x-2
cbe = arcsin
V9-U-2)2
. x-2
dx ,
+ a
V5 + 4 jc - x2
dx = 6 arcsin
x-2
4^5 + 4jc -jc2 + C
Integrating jf(x)f\x) dx
An integral in two factors where one factor is the derivative of
the other is always easy to integrate by the formula
lf(x)]2
\f(x)f\x)dx = UK " +C.
Why? Withf(x) = u, f'(x)dx = du, dx =
du
J fix) fix) dx =
u f\x)
du
f (*)
w
and
fix)
= \u du = —+C =
\f(x)]
+ C
Integrate:
J (x5 + 3 x2 - 7) (5x4 + 6 x) dx
f [tan x (7 sec2 x)] dx
r 4 In*
3 jc
cbe
= - (x5 + 3 x2 - 7)2 + C
= 7jtanx sec2 x dx
4
3
= ^ tan2 jc + C
In jc I — I dr
= o* • In2 jc + C .
J (sin jc) (1 - cos x) dx
2* (1 - cos jc)2 + C .

Section 21.2 Methods of Integration
745
Integration by Partial Fractions
pp. 130 - 31
An algebraic expression containing a polynomial in the
denominator, or in the denominator and numerator, can
always be integrated by splitting the function into partial
fractions, each amenable to a known integration formula.
Integrate
jc2 + x + 1
2 x4 + x3 + 2 x2 + x
dx.
x2 + x + 1
2 x4 + x3 + 2 x2 + x
dx =
x
— dx - —
x 5
2x \
5 (2 x + 1) 5 (x2 + 1) 5 (x2 + 1))
dx
2x \
2x+ 1
With u = 2 x + 1, and du = 2 dx and dx =
x2+l
du
~~2
x2 + 1)
, we have
dx .
p. 723
2x+ 1
dx = 3
- du = 3 In 12 jc + 11 +Ci
u i i x
p. 724
*2+l
cLt = arctan x + C<l .
p. 742
2jc
j^+1
dx = 2
JC
*2+l
cLt = In (x2 + 1) + C3 .
jc2 + jc + 1
2 x4 + x3 + 2 x2 + x
dx
= In I jc I — — [3 In |2jc+1| - arctan jc + In (jc2 + 1)] + C . •
If the degree of the numerator is greater than or equal to the
degree of the denominator, then the denominator must be
divided into the numerator until the degree of the remainder is
less than that of the denominator.
Division: pp. 129 - 30
Integrate
f jc2 + 3
jc - 3
dx
jc2 + 3 12
= jc + 3 +
JC — 0 JC — 0
C x2 + 3 1
dx = ;rjc2 + 3jc+12ln |jc-3| +C
jc- 3 2

746
Chapter 21 INTRODUCTION TO INTEGRAL CALCULUS
Integration by Power Series Expansion
p. 768
p. 771
A function f(x) that can be expanded as a power series in x
may always be integrated if fix) is replaced by the terms of the
series.
Integration by power series expansion is used to evaluate
integrals that can only be expressed by an infinite number of
elementary functions.
The integrals
e~x dx and
sin x
x
dx
cannot be expressed in terms of a finite number of elementary
functions.
Term-by-term integration of Maclaurin's expansion
e - i 1{ +2, ...
gives
f 2 i *^
e~x ax = x -
xK
x
3 1! 5-2! 7 • 3!
+ ... +C.
Using the expansion
s «A/ *A/ «A/
sin* = ^-37 + 57-71-
+ ...
we have
sin x x2 x4 x6
T"~" = 3!" + 5T ~ 1\+'"
and
r sin x , x3
dx = x-
xK
X
X
3-3! 5-5! 7-7!
+ C.

747
21.3 The Definite Integral
Riemann Sum
An integral that is defined between two values a and 6 of an
independent variable is a definite integral.
An indefinite integral \f(x) dx is a function plus an
arbitrary constant; a definite integral
b
J fix) dx
a
is a quantity.
Geometrically, a definite integral can be thought of as the area
contained between the graph of a function and the x-axis of an
orthogonal coordinate system in a closed interval from x = a to
x = 6. Areas located above the x-axis count as positive in
integration, areas below the x-axis as negative.
y =fb)
a=xi
To find an approximation of the area, we divide the interval
[a, 6] into n sub-intervals at the points
with
a = xq<x\<X2 < ... <X{ ... <xn = b
b — a
A x = .
n
Within every sub-interval A a; we choose an abscissa &; the
total area A under the graph is approximated by the sum of the
areas of rectangles, /X&) • Ajcj,
n
A * ^fi&.Axi,
; = i
known as a Riemann sum.
With an increasing number of subdivisions, and forever
smaller sub-intervals Ajcj, the Riemann sum continuously
approaches the sought area more closely. That is,
n
A = lim 2l f(£0 ' Axi •
The definite integral is defined to be this limit:
b n
jf(x)dx= lim 2u f($i) > Axi .
The limit exists whenever f is a continuous function on the
closed interval [a, 6] .

748
Chapter 21 INTRODUCTION TO INTEGRAL CALCULUS
RIEMANN Georg Friedrich
(1826-1866)
Bernhard
The definite integral is sometimes called a Riemann integral
after the German mathematician Bernhard Riemann, whose
work on trigonometric series in 1854 prompted a precise
definition of the integral.
The Fundamental Theorem of Calculus
von LEIBNIZ Gottfried Wilhelm
(1646-1716)
NEWTON Isaac
(1642-1727)
Although some notion of integral calculus is at least 2500 years
old, tangent problems and area problems had no unified
technique for solution as recently as the latter part of the 17th
century - each particular problem involved the employment of
special methods.
A breakthrough in mathematics and physics came in the 1670s
when Leibniz, in Germany, and Newton, in England,
recognized the inverse relationship between the tangent problem
(differentiation) and the area problem (integration),
condensed in the fundamental theorem of calculus. From the
plethora of "infinitesimal techniques" developed in the 16th
century and the first half of the 17th century, Leibniz and
Newton had extracted a powerful system - the infinitesimal
calculus or, with present-day terminology, calculus - a feat
often referred to as their "discovery of calculus".
Let A ix) be the area between the graph of the continuous
function fix), the x-axis, and the verticals through the points
xq = a and x.
The area increment is
A A & f(x) • Ax .
When A x tends to zero, we have the derivative of A (x),
dA A(x + Ax) -A (x) f(x)-Ax
dx~= hm A^ = hm A! =f(xh
ax A x -> 0 aX A x -> 0 a X
that is, fix) is the derivative of the area A ix).
If/is a continuous function in the closed interval [a, b], such
that
Fl ix) = fix) for all x in [a, b] ,
the fundamental theorem of calculus states that the definite
integral
b
jfix) dx = Fib)-Fia) .
a

Section 21.3 The Definite Integral
749
Finding Areas
Substitution Symbols
An important use of definite integrals is the determination of
the area between two curves, generally one curve and the
x-axis.
i b b
For convenience, we use the substitution symbol or [ ] ,
I a a
b
J f(x) dx =
a
b F(x) = [F (x)]b = F(b)-F(a).
a a
A definite integral may be split into two or more integrals of
the same function, thus,
b m b
\f(x) dx = J f (x) dx + \ f (x) dx ,
a
a
m
where a <m <b; with the association of definite integrals with
area, we have
A = Ai +A2.
x
Determine the area contained between the graph of
fix) = x3 + 3 x2 - 10 x
and the x-axis in the interval from x = -6tox = + 4.
Plotting the function gives the result:
/Tx)=*3 + 3x2- IOjc
x

750
Chapter 21 INTRODUCTION TO INTEGRAL CALCULUS
The graph shows that parts of the area are located below the
x-axis and, consequently, will appear as negative quantities
in integration.
The area between x = - 6 and x = + 4 will therefore have to be
integrated separately over the intervals
x = [-6,-5] [- 5, 0] [0, + 2] [+ 2, + 4] .
The indefinite integral is
x4
\ (x3 + 3 x2 - 10 x) d x = — + x3-5x2+C,
-5
-6
F(x) =
o
-5
+ 2
0
Fix) =
which gives
f^-125-
^4
) + 93^
4
125 -
^1296
I 4
=
-216-180 J
-93|+ 72 =
4
=
integral
-»!
3
+ 93T
4
area
»!
931
4
Fix) =
^T +8-20 1-0 = -8-0
V4
-8
8
+ 4
+ 2
Fix) =
(256
+ 64-80 -(-8) = +48 + 8
= +56
56
120
179j
Changing Limits
An interchange of limits reverses the sign of the definite
integral; thus,
b a
\fix) dx = - J f ix) dx .
a
When a definite integral has been evaluated by the method of
substitution of a variable, it is generally more convenient to
use the limits of the new variable than to convert back to the
original variable.
V2
Find
V4 - x'
dx .
j
1
X'
V4 - x'
dx =
f V22-*2
dx
X'
XA

Section 21.3 The Definite Integral
751
p. 738
We have already found by substituting
x = 2sin0; dr = 2cos0d0
that
y±-x'
xA
dx = - cot 0- 0+ C.
Instead of reinstating the original variable x, we determine
the limits of the new variable 0.
x = 1
x = ^2
K
2 sin 0=1; 0 = -
2 sin 6 = V2 ; 0 = j
Thus,
V2
dx =
j
1
xl
k/4 / n
(- cot 6 - 0) = I -1 - T
n/6
W3 -|
= (^3-1)-^5-= 0.47025...
12
Integrated Mean Values
y =fix)
X
The mean-value theorem for integrals states that:
If/"is a continuous function in the interval [a, 6], then
there exists a number c between a and 6 such that
\f{x) dx = (b-a)-f(c) .
a
The average value of a finite quantity of numbers a\, a<i... an is
the arithmetic mean
n
Mn =
The value f(c), as defined by the mean-value theorem, is
referred to as the average value offix) in [a, 6].

752
Chapter 21 INTRODUCTION TO INTEGRAL CALCULUS
To see why f(c) can be thought of as an average value, divide
the interval into n equal sub-intervals, each with a length
b — a
Ax =
n
between the points of division
•^1 *^2 • • • **71 >
the average value of the midpoints /X£i), (¾) ••• (%n) of the sub-
intervals Axi,Ax2 ...Axn is
f(tl)+f(t2)+-+f(tn)
n
or, since n =
b —a
Ax '
n
jT7 mi)+a&) + -+ mn)] a* = ^1 fito a *.
i = 1
When n tends to infinity, we have
1 n 1 b
lim T X/X^)A:x; = I \ fix) dx.
b — a ' b — a i
"i = l " ** a
n —> oo
To summarize, the average value f(c) in [a, 6] of a continuous
function f(x) is determined by the formula
/(c) =
b — a
J f(x) dx
a
Find the average value of
fix) = 1 + 3 # - x2
in the interval [-1, 2] .
12 i r 3 1 ~|2
f(c) = o /_■. v J (1 + 3 x -X2) dr = - * + - x2 - -x3
3 l2J 2 '
-= 1 + Sc-c2
c = -(3 + V7), discarded ; c = -(3 - V7) ** 0.177
x

753
21.4 Multiple Integrals
An integral in which the integrand is integrated twice is a
double integral,
\\f(x) dx2;
if three times, a triple integral,
J \\f(x) d*3;
etc., to an n-fold iterated integral,
J J ...\f(x) dxn .
The integration is performed from the inside out.
3 2
Find J J 2x dx2 .
-1 o
\ 2x 6x = x2 + C\ ;
\ 4 dx = 4x + C2 ;
Hence.
x2 = 4.
0
4jc = 16
-1
3 2
J J 2x dx2
= 16.
•1 0
Functions of more than one variable may be integrated with
respect to one variable at a time while the other variables are
held constant, reversing the process of partial differentiation.
3 2
Find
0
J (4 y3 + 2 x) dx dy
Hold y constant and integrate with respect to x:
\ (4 y3 + 2 x) dx = 4y3 x + x2 + C ;
[4y3x + x2]X = 2 = (4y3.2 + 22)-(4y3 +12) = 4y3 + 3
x= 1
Integrate with respect to y:
J(4y3 + 3)dy =y4 + 3y + C;
\y4 + 3 y ] = 81 + 9 = 90 .
0
Hence,
3 2
J J (4y3 + 2jc) dr dy = 90 .
0 1

754
Chapter 21 INTRODUCTION TO INTEGRAL CALCULUS
2 14
Find J J J (2 x + 4 y - z) dx dy dz .
-1 0 1
2 14
2 1
J J J(2x + 4y-z)dxd<yd2:= J J [x2 + 4:ry - xz] dy dz
-1 0 1 -10 x
2 1 2 x
= J J (12y-3z+ 15) dy dz = j [6 y2 - 3y z + 15 y] dz
-1
2
= J (21-3z) dz
-1
212-^22
2 1
-1 z
A limit of integration may contain a variable of the integrand
3 2y
Find J J Sx2 dx dy .
l 0
3 2y
j j Sx2 dx dy = j [x*f 2y dy = j 8y* dy =
1 0
x = 0
2 v4 = 160
Geometrical Interpretation of the Double Integral
We are familiar with the geometrical interpretation of the
equation y =f(x) as a curve in the two-dimensional #,y-plane,
b
and of the integral J f(x) dx as an area between the curve and
the x-axis. a
Similarly, while the equation z -f{x,y) defines a surface in
the three-dimensional #,^,2-space, the double integral of a
b d
continuous function of two variables, J J f(x,y) dx dy, may
a c
be interpreted as a volume between the surface z = f(x, y) and
the #,y-plane.
z
x
&yi
'4 Ax;
AAj = Axi Ayi

Section 21.4 Multiple Integrals
755
In the graph, the rectangular area AA; in the x, y-plane is
projected on the surface z = f(x, y).
The quantity AA; = Ax; Ay; is the area of the bottom surface of
a column whose top surface is part of the surface /"(*;, y;).
The smaller the area AA;, the closer the volume of the column
is to that of a parallelepiped measuring f{xi, y;) AA;.
If the domain Z), consisting of {x, y) with c < x < d, a <y < 6, is
divided into an increasingly greater number of rectangles
so that AA; tends to 0, then the volume between the surface
z =f(x,y) and D equals the sum of all parallelepipeds
measuring f(xi9yi) Ax; Ay;; thus,
* b d
V= lim ^f(xi,yi)AXi Ay. = \ \ f(x,y) dx dy.
/i —* °° j = 1 a c

756 Chapter 21 INTRODUCTION TO INTEGRAL CALCULUS
21.5 Improper or Unrestricted Integrals
Up to this point, we have discussed definite integrals, also
referred to as Riemann integrals, which are guaranteed to
exist only for continuous functions on closed intervals.
However, certain problems create a need for an integrand that
is infinite within the range of integration or an integral on an
infinite range of integration; such integrals are generally
referred to as improper integrals.
There is nothing "improper" about so-called improper
integrals other than not being covered by the definition of the
definite integral, and the term improper is rather unfortunate
and misleading.
The French integrate impropre is equally infelicitous, less
so the German uneigentliches Integral, while the Swedish
term generaliserad integral avoids the negative connotation
entirely.
Since the improper integral has no restriction as to interval or
function - other than being unrestricted - we suggest that
Unrestricted Integral unrestricted integral might be a more suitable term.
Evaluating an unrestricted integral will always involve
calculating a definite integral and a limit.
Infinite Integrands
This group of unrestricted (improper) integrals has
integrands that become infinite in the range of a finite interval of
integration.
Using the terminology from infinite series, we say that the
integral converges if the limit exists as a finite number. On
the other hand, if the limit does not exist, then the integral
diverges; a diverging integral may be said not to exist.
We consider the following three typical possibilities for
infinite integrands:
1. If f is continuous in ]a, 6] and has a vertical asymptote
at a (that is, f becomes infinite at a), then
b b
\f(x) dx = lim J f (x) dx,
a c —»a + c
provided that the limit exists as a finite number.
2. Analogously, if f is continuous in [a, b[ and has a vertical
asymptote at b (that is, f becomes infinite at 6), then
b c
\f(x) dx = lim jf (x) dx,
a c —> b~ a
provided that the limit exists as a finite number.

Section 21.5 Improper or Unrestricted Integrals
757
3o If f is continuous in [a, b], with the exception of a point
c with a < c < b, where f has a vertical asymptote (that is,
f is infinite at c), then
b c b
jf(x) dx = jf(x) dx + \ f (x) dx
a
a
provided that \ f (x) dx and j fix) dx exist.
a
In agreement with the use of the term unrestricted integral, it
is appropriate to say that type 1 is "unrestricted in a", type 2
"unrestricted in 6 ", and type 3 "unrestricted in c " .
Find
1
r 1
x
1/2
dx.
o
The integrand becomes infinite at x
We have
= 0.
l
r 1
x
1/2
dx = lim
1
r 1
0
c->0+ J
c
X
1/2
dx
Solve in two stages:
1:
^ 1
x
1/2
dx = 2xy2 + C
= 2V* + C,
l
^ 1
JC
1/2
dx =
2-[x
= 2-2a/c
2:
lim
c->0 +
= lim
c->0+
JC
1/2
dx
2-2a/c = 2-0 = 2
units
= v-1/2
JC
Thus, the integral converges, and
l
r 1
o
X
1/2
dr = 2
The area bounded by the curve of y = x~1/2 and the x-axis from
jt = 0tojt=lis2 square units.

758
Chapter 21 INTRODUCTION TO INTEGRAL CALCULUS
Find
1
dx
0
The integrand becomes infinite at x = 0.
j
o
l
x<
dx = lim
c->0+ J
c
1
r -L
dx .
Solve in two stages:
1:
r 1 . 1
1
'•J.
X4
n dx = -— +C,
2 jc
dr =
x c
2: lim
c->0+ J
1
X5
dx = lim — - 1 =
oo
1 =
oo
C->0
+ \c
Thus, the integral diverges, and the integral
j
o
l
x<
dx is
infinite; and the area bounded by the curve of y = —- and the
x-axis from x = 0 to x = 1 is infinite.
Find
5
r 1
j
0
x - 1
dx.
JC
This example calls attention to the need for assessing an
integrand for asymptotes before evaluating the integral, so
that an unrestricted integral is solved in terms of limits and
not confused with a definite integral.
The integrand becomes infinite at x = 1.
We have
j
o
5
r 1
x —
dx =
j
o
l
r 1
x - 1
dx +
x - 1
dx .

Section 21.5 Improper or Unrestricted Integrals
759
Evaluate the components, one at a time, by a two-stage
approach.
1
r 1
x - 1
dx =
0
lim
x - 1
dx
0
1:
=- dr = In I jc — 1 I + C ;
c
r 1
jc — 1
dr =
0
0
In I jc — 1 | = In | c - 11 , for c < 1
2: lim
c-> l-
0
x — 1
dr= lim ln|c-l
C-4 I-
^ — oo
The nonexistence of a limit at one endpoint is sufficient to
manifest the divergence of the integral; as the first component
integral diverges, there is no need to evaluate the second,
5
— dx.
x — 1
Thus,
ofy =
5
r 1
x — 1
d x diverges, and the area bounded by the curve
o
X - 1
and the x-axis from x = 0 to x = 5 is infinite.
x
i
i

Chapter 21 INTRODUCTION TO INTEGRAL CALCULUS
Integrals with Infinite Limits
Assume that f is continuous in the half-open interval [ a, <*>[;
then, for every finite value b > a, we have a definite integral
b
F(b) = jf(x) dx.
a
If b tends to infinity, we have the unrestricted (improper)
integral
oo b
j fix) dx = lim jf(x)dx.
a b —><» a
y
>* X
a b
The integral over an infinite interval exists only if the limit
exists, and it is then said to converge; if the limit does not exist
as a finite number, the integral is said to diverge.
We have the following possibilities for unrestricted integrals
of infinite limits:
1. If/"is continuous in [a, ©o[,then
oo b
\f(x) dx = lim jf(x) dx,
a b —><» a
provided that the limit exists as a finite number.
2. If f is continuous in ] -<», b], then
6 6
J fix) dx = lim jf(x) dx,
— oo a —> — oo a
provided that the limit exists as a finite number.
3. If/*is continuous in ]-<», ©o[ and c is any real number, then
oo c oo
\f{x) dx = jf(x) dx + jf(x) dx,
— oo — oo C
provided that both integrals to the right of the equality sign
converge, in which case the sum on the right is
independent of c.
It is appropriate to say that type 1 is "unrestricted in <*>", type 2 is
"unrestricted in -oo", and type 3 is "unrestricted in -<» and ©o".

Section 21.5 Improper or Unrestricted Integrals
761
Find
dx
X'
r l
X'
dx = lim
b —>oo ,,
1
b
r j_
X*
dx
1:
*2
6
x<
dx =
dx =
x
b 1
-=1
1 X
1
6
2: lim
b —>oo ,/
1
6
JC5
dr = lim
6 —>oo
1
b
1 - f =1-- =1
oo
The integral converges, and
dx = 1
X'
The area bounded by the graph of y = 1/x2 and the x-axis from
jc=ltox = ooisl square unit.
y
3"
1-
y =
1/x2
\ A
]
L
oo
1
3
= 1 sq. unit
\
oo
X
Find
— dx
x
— dr
= lim
b —>oo J
1
b
— dx
1:
— dr
= In x + C ;
— dr
In x = In 6

762
Chapter 21 INTRODUCTION TO INTEGRAL CALCULUS
2: lim
b
b —>°o J
— dx
x
= lim ln6 =
b —>°°
oo
Thus, the integral diverges, and the area bounded by the curve
of y = — and the x-axis from x = 1 to x = °° is infinite.
x
oo
X
Find
l
/• 1
x*+l
dx.
We have
*2+l
dx = lim
a —> — oo J
a
X*+l
dx
p. 724 1:
*2+l
dr = arctan x + C ;
l
' 1
a
*2+l
dx =
arctan jc =
f
a
K ,
t - arctan a .
V
2: lim
l
r i
K
a —> — oo j
a
n -dr=-r- lim arctan a
x2+l 4
a —> — oo
7C
4
2
V
4 '
Hence,
l
r 1
./
— oo
37C
— dx = —7- .
x2+ 1 4

Section 21.5 Improper or Unrestricted Integrals
763
— oo
a
= 3tc/4 sq. units
x
Find
I x e~x'
dx.
x e~~x dx = \x e"^ dx + \x e~x'
dx.
Choose c = 0 .
Evaluate the components, one at a time, by the two-stage
approach.
0
x e~x
o
dx = lim I x e~x dx .
p. 734 1: [x e-*2 dx = -1 e"*2 + C ;
0
| x e~~x2 dx = - — e"*2 = --(^
a
1-e
-a
!).
0
2: lim x e~^ dx
a -> -°° a
lim -|(l-e-«2)
Since the first component integral converges, proceed to the
next component,
oo 6
I x e~x
o
dx = lim x e~x dx.
6 —>00 Q
: x e~~x*
dx =
-1 e-*2 + C
I x e~x*
o
dx = - —
1
2
e-x —
0
-|(^-i)

764
Chapter 21 INTRODUCTION TO INTEGRAL CALCULUS
2: lim x e~x dx
b —>°° o
lim-i(e-62-l)
6->~ z
= -|(0-l) = |.
Thus, both component integrals converge, and
j
x e~~°^ cbc = -— + — =0
— oo
oo
The area bounded by the curve of y = x e~x and the x-axis is
1_
2
+ -=1 square unit.

765
Chapter
22
POWER SERIES
Page
*** 8 Introduction to Sequences and Series 263
22.1 Convergence 766
22.2 Taylor's and Maclaurin's Series 767
22.3 Expanding Transcendental Functions 771
22.4 Binomial Expansion 775
22.5 The Riemann Zeta Function and Hypothesis 778
Indicates cross-reference

766
Chapter 22 POWER SERIES
22.1 Convergence
A power series in ascending powers of x may be written
oo
^CLkXk= (20 + (21^1 + CL2X2 + CL3X3 + ... ,
k = 0
where by convention the 0-th term is taken to be a 0 even if x = 0,
when x° would ordinarily be undefined. More generally, we
may seek an expansion of a function fix) in powers of ix - c),
00
fix) = ^akix-c)k= lim a,Q +aiix-c)1 + a,2ix-c)2 + ... + a nix-c) n ;
k = 0
n —><»
if this limit exists, it defines the sum given. Here c is the
center of convergence. In the figure below, R denotes the
radius of convergence; the interval of convergence is the
interval that consists of all values of x for which the series
converges, that is, for which the limit exists:
fix)
A
interval of
convergence
x
A power series centered at c converges for
\x-c I <R
and diverges for
\x-cI >R .
If the radius of convergence is 0 CR = 0), then the series
diverges whenever x * c.
If R = 00, then the series converges for all x.

767
22.2 Taylor's and Maclaurin's Series
A function f that can be expressed as a power series centered at
c may be written
fix) = ao + a\(x- c) + a<i(x - c)2 + a^ix — c)^ +... .
How might the coefficients ao, aj, a^ ... be determined?
To find a0, we evaluate /(c), because all terms but the first
disappear when x = c:
f (c) = a o + #1 (C - c) + (22 (C - c)2 + (23 (c - c)3 + ...
f (0 -, ,
What about ai? Assuming that term-by-term differentiation is
justified, we find the derivative f ' (jc); as in the previous step,
all terms but the first vanish for x = c:
f\x) = CL\ + 2 (22 (X - C) + 3 (22 (X - C) 2 + ...
/ '(c) = a\ + 2 a2 (c - c) + 3 a<i (c - c)2 + ...
f • (c)
/"(c) = ax or ax = —=-j—= f ' (c).
Continuing with /*", we get
/""to = 2a2 + 6a3(x-c) +...,
which gives
f (c) = 2 a2 or a2 = —^-j—.
Continued term-by-term differentiation followed by
substitution of x with c, and solving for a n, gives
Gn " k ! '
where&! = 1 • 2 • 3 • ... • (k - l)k.
Thus, if f(x) is represented by a power series
f(x) = a o + (2i (x - c) + a2 (x - c)2 + a3 (jc - c)3 + ...
where c denotes the center of convergence, and term-by-term
differentiation is justified, then
oo
fix) = 2_. "~^~i— (x - c)R
fc = o
(x — c)2 (x — c)3
= /(c) + /'(c)(x-c) + /"(c) 2! + /'"(c) 3, + ■■■
in the interval (- R + c, i? + c), where R denotes the radius of
convergence.

768
Chapter 22 POWER SERIES
TAYLOR Brook
(1685-1731)
GREGORIE James
(1638-1675)
This power series is Taylor's series, named after the English
mathematician Brook Taylor and published in his Methodus
incrementorum directa et inversa (1715).
Although first published by Taylor, this series had been known
some 40 years earlier, as evidenced by unpublished papers of
the Scottish mathematician and astronomer James Gregorie.
METHODUS
Incrementorum
DireQa & Inverfa.
AUCTOKE
BROOK TAILOR LL.D. &
Regit* Socieutu Sccrctario.
I 0 N D 1 N I
Typ'u Pt*rfomi*M* : Profhat«apud Gml. U*}i ad Infigoia
Principit in Csmctcrto Pjuiiino. MDCCXV.
( « )
DEMONSTRATIO.
»
X^IX-^*
• ••
x + 4 x-\- 6 x-¥ 4 x-f x
• ••
x-f x
• • •
x-{-2 X-f *
• it
X
x-\-x
•• • •
x+ix + x
< « • t
« *•
x-fx
v :i
I
X
&C.
J*C
Vaforcs fucceflivi ipfius x pa additlonem contiouam colleCH font
x, x+x, x+2x-fx, x+^x+jx+x, ficc. ut paict pa operatlooea
•
in tabula anncxa expfcAam. Scd in his valoxibos x CQcffidentcs
numerates terrainoram x, ut x% ficc. eodem roodo formamur. ac
cocfficientes terminorum conefpoodcntium in dignitate binbmlL

Et (per Theorem* Nevtonumtm) fi dignitatis index fit», coeffid*
769
gd quo tempore t aefcendo fit % + *%, hoc eft x -f »f fiet» zqot-
■ • 1 « 7 1 a * ■» +* •*
s«d««-- ^ .p)_«. _■ ^^ -)i.ss-
/«*--st \ ^^ Kc Ptolnde qpo temporex cteTcendofitx+v,
eodem tempore x crefcendo Met x >\> « L. -f. » ** J4-r «po.
C 0 R O L L. I.
Et ipfis z, x, x, .*,&c.iirdemmanentH>us, routato figeo ipfiusv,
quo tempore x decrefcendo fit * — v, eodem tempore x decrefcen-
do fiet x — x -s+ x -J2L-. — *.. »"" , &c. vd juita notit?©.
4 !* -* 1.six1 v 1.2. 3^« J
cy v 0 0
nem noftram x — »-- 4 x 1_ — x " ace, Ipfii *,«,&&
' . "1.2ft* V I.2.JXI
converts In — vy — o, ficc.
COROLL. H
8} pro Increment)* etaneiccnribin ftrJbanror ftuadorn* \tfm ft*.
pottlaatlM, ftOb Jam onotlps ^ ^% u, v. ^ tyriUm
* A
«1 ^
quottmponsx tmlfbtttHwifr^illb fit k*tr fct*, *Hh«4 +
« -1*- + * , „M ** wiomW %» Wf«* v, quo tcm-
1. W ,-»-ST
port xdecxtfcendo fit * — % x decrtfctfv5o ftt x - i i~- *
"Taylor's series" as described (this page and opposite page) in
Methodus incrementorum ... (1715). Taylor used a system of dots (for
increments) and superscript/subscript primes, to which the modern
reader is usually unaccustomed.
If centered at 0 (that is, c = 0), the Taylor series is known as a
Maclaurin's Series Maclaurin series,
00
,(,,.^2^0),.
k !
k = 0
= /(0) + f'(0)x +—9~j— +—3!— +

770
Chapter 22 POWER SERIES
TREATISE
o t
FLUXIONS.
la Two BOOKS,
B Y
Colin Mac Laurie, A. M.
Pnftfftr $f Mathematics in* the Onrvajhy *f
Edinburgh, and FtUnv *f tht Royal Society*
VOLUME I.
E'DINBVRGHi
Printed by T. V» and T. Ruddihak*.
Moccxui*
MACLAURIN Colin
(1698-1746)
STIRLING James
(1692 -1770)
In his Treatise of Fluxions (1742), Colin Maclaurin described
Taylor's series in a manner still respected in today's
textbooks. Maclaurin made no pretense of having discovered the
series that now bears his name — on the contrary, he referred
to it as a special case of Taylor's series. And even before
Maclaurin's treatise, Maclaurin's series had been used by the
Scottish mathematician James Stirling.
Taylor's Formula with Remainder and Error
Estimate
Taylor's formula with remainder approximates functions b]
polynomials and provides estimates for errors.
If a function f(x) has term-by-term derivatives f^k\x) in th(
closed interval [- R + c, R + c], where R denotes the radius o
convergence, Taylor's formula with remainder states that
(x - c)2
fix) = f(c) + f'(c)(x-c) + f"(c)
+ f" (c)-^...+ f(k-1](c) (X
2 !
\k
3 !
(k - 1) !
+ remainder.
The remainder term may be determined, but we leave the
discussion of the various methods out of our discussion here.

771
22.3 Expanding Transcendental Functions
Except for special triangles, we find numerical values of
trigonometric functions by trigonometric tables or a
calculator. Similarly, tables and the calculator are used to find
values of natural logarithms. Power series expansion helps
with the calculation of such functions. The values are
approximate, but so are those given in tables and by the calculator.
The first term of a converging series is the first
approximation, the sum of the first and second terms the next
approximation; by adding an increasing number of terms, the
degree of accuracy is limited only by our finite capacity of
adding terms.
To form polynomials from the terms of a convergent series,
Taylor's formula with remainder is the concluding - but not
easily applied - tool for approximation.
Some series lend themselves to convenient error estimation.
Thus, if for a particular x the terms of the series alternate in
sign and decrease in absolute value, the absolute value error
introduced when approximating the function at x by the sum of
terms from the beginning of the series cannot exceed the
absolute value of the first term discarded.
Expanding sine of x
sin x-x
•A/ «A/
37+ 57
x'
77
where x is in radians.
Why? Assuming that sin x can be expressed as a Maclaurin series,
we determine the coefficients of the series:
Let f(x) = sin x, where x is in radians.
sin 0 = 0; cos 0=1
fix) =
f ■(*) =
f"W =-
f'"ix) =-
sin x => fiO) =
cos x =» /"' (0) =
-sin* => f'iO) =
-cos* => f" (0) =-
Using these results ]
fix) = fi0)+f
we obtain
fix) = sinx= 0
0
1
0
1
fM ix) =
sin x => j
/■(5) ix) = COS X => i
f(®(x) =- sin x => i
/*(?) ix) =- cos X => i
otn
in Maclaurin's series
(0) x + f xL + f
1 T^ \ T^
+ X
+ 0 +
I 3lJ
+ 0 + —+0 +
5 !
f (4) (0)
f (5) (0)
r(6) (o)
fM (0)
•Ar ^9 • • •
( *^
l 7!j
= 0
= 1
= 0
= -1
>
+ ...
XK
X'
X

772
Chapter 22 POWER SERIES
y=X-X3/3l »
i
y = sin x
y =
= x - x3/3\ + x5/5l - x7/7l
x
y = x- x3/3! + x5/5l
The graph shows improved agreement between the graph of
sinx and graphs of polynomials containing an increasing
number of terms obtained by power series expansion of the sine
function.
Since the terms of the sine series are of alternating sign and
steadily decreasing magnitude for \x\ < 1, the error
introduced by breaking off the power series at any term will not
exceed in magnitude the value of the first term left behind.
Employ the first eight terms of the Maclaurin series expansion
to express the value of
sin 0.2 rad .
Since
sinx & x -
yd yu
•A' •A'
3T +FT
X
xc
X'
X
we obtain
sin 0.2 rad * 0.2
7 ! ~ X 6 + 120
0.23 0.25 0.27
—7.— +
5040 '
120 5040
Find the first four terms of the Maclaurin series of
y = sin (2 x + n) .
fix) = sin(2x + n)
f ' (x) = 2 cos (2x + n)
f" (x) = -4sin(2x + 7c)
f '" (x) = -Scos(2x + n)
/(0) = sin n = 0
f'(0) = 2cos7c =-2
/"'(0) = -4 sinn = 0
/'" (0) = -8cos7c = 8
fix) = /(0)+/'(0)x+-^P X2+I—B. j3 + _
8 o
= 0 + (-2x) + 0 + -r-rx3
4 x3
& —z— - 2x .

Section 22.3 Expanding Transcendental Functions 773
Expanding cosine of x
•%* •%* •%*
COS* * 1-—+—-—+ ...,
where x is in radians.
Why? Assuming that cos x can be expressed as a Maclaurin series,
we determine the coefficients of the series:
sin 0 = 0 and cos 0 = 1.
f{x) = COS X =>
f ' (x) = - sin x =>
f " (x) = - cos x =>
f "' (x) = sin x =>
/*(4) (x) = COS X =>
/*(5) (x) - - sin x =>
/*(6) (x) = - COS X =>
etc.
Using these results in Maclaurin's series
f(x) = /(0) + /" (0)*+ f2 [0) *2+ ^ !(0) ^3+...,
/(0)
/'(0) =
/•M(0) =-
/•"•(0) =
/(4)(0) =
f® (0) =
/(6)(0) =-
1
0
-1
0
1
0
-1
we obtain
/"to = COS* = 1 + 0 - ttt + 0 + — + 0 - TT +
2 ! 4 ! 6 !
•%* •%* •%*
2~r+IT- 6T+ ••••
Expanding Natural Logarithms
i , ^ fa -1)2 fa -1)3 fa -1)4
In x = (x -1)- - + 3 " 4 +
... + (-lr —k - i—+ ' —5£—
where x is a real number with I x - 11 < 1 .
Why? The function In x is not defined for x = 0, so any Taylor series
expansion will have to be centered at some point c > 0. The
choice c = 1 is conducive to convenient differentiation:
Employ Taylor's series; let c = 1:
f" (1)
fix) = In* = /(1) + /"(Dfa-D+ 72! fa-D2
+ —g-j— fa- lr +....

774 Chapter 22 POWER SERIES
fix) = lnx = 0+ (!)(*-!) +
fix) = lnx ==> fil) =lnl = 0 (e° = 1)
T(l) =1
/"(1) =-1
/'" (1) =2
/-(4)(1) = -6
Using the above coefficients,
(-1) ix - 1)2 2 ix - 1)3 (-6) ix - 1)4
f 'fa)
f" fa)
f -fa)
f(4) fa)
etc.
1
= —JC
= 2jt
= -6.
-2
-3
X~
4
2 ! 3 ! 4 !
(*-l)2 (a: - 1)3 _ (* - 1)4
^X J. y ~ + ^ . +...,
which is valid for 0 < x < 2.
Employ the first four terms of a power series expansion of In x
to determine the numerical value of In 1.3.
< ^ fa ~ I)2 fa ~ I)3 fa - I)4
In x & (x - 1) — - + - r .
Discarding the last term of the polynomial, we obtain
! ,o /, o ,x (13-1)2 (1.3-1)3 __
In 1.3 * (1.3-1)--1—-——+-—-—-=0.264.
The terms of the series alternate in sign steadily and decrease
in absolute value, so the limit lies trapped between any two
successive sums as follows:
(1 3 - 1)4
0.264 < In 1.3 < 0.264 + -1—-—- = 0.264 + 0.002 025 .
4
So, to two decimal places
In 1.3 = 0.26... .
Expanding ex
The base of the natural system of logarithms is defined
e = lim
n —» oo
l + — ;
it can also be expressed as the sum of an infinite series,
e = lim
n —» °°
\ J_ J- J- J-
^ 1 ! + 2 ! + 3 ! + '"+ n !
To expand e*, employ Maclaurin's series,
fix) = fiO) + —j"j— + -^g]— x +—3I ^ + ••• + >
and let fix) = ex.

Section 22.4 Binomial Expansion 775
p. Ill As every derivative of e* is e*, and e° = 1,
AO), f' (0), f " (0), f'" (0) ... must all equal 1;
and assuming that e* can be expressed as a Maclaurin series,
we have / 2 -.3 y/i\
e* = lim 1+77+777+:77+...+ —r .
n-> <*> V l!2!3! n \)
Alternatively, the series expansion for e* can be approached
( x\n
using the binomial theorem and e*= lim 1 + — .
n —> oo y n J
22.4 Binomial Expansion
p. 140 The binomial theorem, previously discussed in Chapter 4, says
that
(X+y)P = XP + pxP-ly + P(P~1).Xp-2 2+ PtP-1)^-^) p.3 3 +
-7 ^ -7 12 1-2-3
p(p- l)(p -2)... (p-tt+ 1) ,
... + - - -xP~nyn+ pxyP *■ + ...+yP,
1 • 2 -3 ...rc J H J J
wherep is a positive integer, and n = 0, 1, 2 ... p may be proved
by direct combinatorical arguments.
The general term, the (n +l)-th term, is
p(p-l)(p-2)...(p-rc + 1)
n
i
xP~n yn ,
p!
pp.196-97 or ___. xP-n yn,
or, in notation of the binomial coefficient, ( ) • xP~n yn .
To find the coefficients of the binomial theorem, we may also
expand f(x) = (a + x)P into a power series, centered at 0:
f(x) = (a+x)P= /-(0) + f'(0)x + f2 \0) x2 + fs t(0) jc3+...
Determine the coefficients:
= (a+x)^ => /(0) = aP
= p(a + x)P~1 => /"(0) ^a^"1
) = p(p-l)(a+x)^-2 => f"(0) =p(p-l)aP"2
• • •
) = p(p-l)...(p-n+l)(a + x)P-n => f(nH0) = p ^-1)...^-^ + 1)(^-^
• • •
) = p(p-l)... (2)(l)(a+x)P-P => f^(0) = p! [(a+x)° = l]
) = 0; n >p+l
For a positive integer p, the series is thus reduced to a finite
polynomial - and we again have the binomial theorem:
(x +y)P = xP + pxP-iy + P(P"1)-^-2 y2 + P(p-D(P-2) 3 3 +
-7 ^ -7 1-2 1-2-3
p (p - 1) (p - 2)... (p -n + 1) „ „ „ n -,
... + -^-^ — —— L-xP-nyn+ ... + pxP-1 + yP
12-3...rc

776
Chapter 22 POWER SERIES
Binomial expansion had been known only for natural number
NEWTON Isaac exponents p - the binomial theorem - when Newton dis-
(1643-1727) covered, in 1664 or 1665, that (a + x)P expands as an infinite
power series in x ifp is a real number but not a natural number
or zero. Ifp is a natural number, the series reduces to the finite
polynomial of the binomial theorem.
As so often happened in Newton's career, the disclosure came
several years after the accomplishment. Only in 1676 did
Newton communicate his discovery in two letters directed, via
the secretary of the Royal Society, to Leibniz.
The power series of
oo
f(x) = (l+x)P =
n = 0
where
(p\ p(p-l)(p-2)...(p-n + l)
\n/ " n !
and/? is any real number except a positive integer or 0,
converges to f (x) for all real numbers x, such that -1 < x < 1
(that is, the radius of convergence = 1).
The proof of the convergence of Newton's binomial series to
(1 + x)P was presented in 1826 - 150 years after Newton's
disclosure - by the Norwegian mathematician Niels Abel,
ABEL Niels Henrik wno investigated the convergence for all real and complex
(1802 -1829) values of the exponent.
Finding Roots
By expanding a radical expression into a binomial series,
we may determine the numerical value through direct
calculation; adding an increasing number of terms, the degree of
accuracy is limited only by our finite capacity of adding
terms.
• To determine a numerical value of
use a seven-term Maclaurin expansion of V a +x .
With f (x) = (a + x)1/2, expand the binomial into an infinite
power series. To do so, first determine the coefficients:
fix) = (a + x)1/2
f (*) =-^(a+x)-1/2
/"' (x) = -j (a + x)-3/2
xe)*».
/(0)
/'(0)
1
2 Va
f'(0) = -
4 a ya

Section 22.4 Binomial Expansion
777
f -(jc) =-(a + x)-5/2
f (5)(*) =^(a+x)-9/2
f "(0) =
fix) = Va +
JC
f (6) (x) =
1 (x2
32
945
64
/*(4) (0) = -
f (5> (0) =
8a2Va
15
16 a3 Va
105
(a+x)-11/2
f (6) (0) =
32a4Va
945
64 a5 Va
fx*
2Va 4aVaV27 8 a2 Va \3 !
15
6t4
= Va
JC
JT
16 a3 Va
X'
105 6c5
945
^t6
V4 V ' 32 a4 Va V5 IJ 64 a5 Va V6 !
5 jc4 7 jc5 21 jc6
2Va 8aVa 16 a2 Va 128 a3 Va 256 a4 Va 1024 a5 Va
With a = 1 and x = 0.3, add the first six terms:
nr^ n—ttz , 0.3 0.32 0.33 5-0.34 7-0.35
Vl.3 = Vl + 0.3 * 1 + — _—+ — __^_+__
* 1.140187 5
The value of the seventh term is
21 (0.36)
1024
= 0.000 014
Hence,
VT3 = 1.1401.
Determine the numerical value of
V6
to three reliable decimals.
Use the series expansion of (a + x)1/2 as in the preceding
example.
Letting a = 4 and x = 2, we add the first six terms of the series:
V4 + 2 = (4 + 2)1^ * 2 +
22
23
2-2 8-4-2 16 - 42 - 2
5-24 7-25
+ -— r—r * 2.44995
128 - 43 - 2 256 - 44 - 2
The seventh term is
21 (26)
1024 (45) 2
= -0.000 64...
Hence,
V6 * 2.449 .

778
Chapter 22 POWER SERIES
22.5 The Riemann Zeta Function and Hypothesis
EULER Leonhard
(1707-1783)
Euler introduced, in 1740, the zeta function as an infinite
series
oo
C(s) =
J_ J_ J_ _ X1 —
" 2s + 3s + 4s +*" " 2^ nS
/i = l
Euler Product
convergent for all real numbers s > 1.
Euler showed that f(s) can also be expressed by a convergent
infinite product, now known as the Euler product,
oo
C(s) =
i-J- i-i. i-i i-i i-L
2s 3s 5s 7s 21s
RIEMANN
Georg Friedrich Bernhard
(1826 -1866)
Refe): p. 87
p. 80
HARDY Godfrey Harold
(1877 -1947)
Im
ttttfltiwm)Mttllll4l|l|MHI"ll!lll|^|il
0 1/2
-►Re
n
/i = l
A
V
P/i'
L^"1
where all values /?„ run over all prime numbers. We have
00
00
C(«) =
5>n
/i = i
^ Pn
- 1
/i= 1
The German mathematician Bernhard Riemann, a pioneer
of modern mathematics, treated, in 1859, f as a function of
a complex variable z; for this reason, f(z) is known as the
Riemann zeta function.
The function f(z) has no zeros in Re(z) > 1; its only zeros in
Re(z) < 0 are at z = - 2, - 4, - 6, ... ; it has infinitely many zeros
in 0 < Re(z) < 1, which are called nontrivial zeros.
Riemann conjectured, in 1859, that all the nontrivial zeros of
f(z) lie on the line Re(z) = ^-, known as the Riemann
hypothesis.
A crucial part of Hadamard's and de la Vallee-Poussin's
proofs (1896) of the prime number theorem was to show that
f(z) * 0 for Re(z) = 1.
In 1914, the eminent English number theorist Godfrey Hardy
proved that an infinity of zeros of f (z) lie on Re(z) = q" , ^u^
beyond that, and the fact that it has been shown that the first
1.5 • 109 zeros in 0 < Re(z) < 1 are all nontrivial zeros on
Re(z) = -r, the Riemann hypothesis has been neither proved nor
disproved.
Since Fermat's last theorem has now been proved, finding a
proof of the Riemann hypothesis might be the most sought-after
accomplishment in number theory.

779
Chapter
23
INDETERMINATE LIMITS
Page
23.1 A Retrospect 781
23.2 L'Hospital's Rule 782

780
FBINCIPIORUM
CALCULI DIFFERENTIALIS
ET
ItfTEGRALIS
EXPOSITIO ELEMENTARIS
AD NORHAM DISSERTATIONS A3 ACADEMtA SCTEWT, REG.
FRU5SICA ANNO 171«. PtL£MU HOrtOKE DECOCUTiS
ILABOaATA
AlltTPlf
SIMONE UHUILIER
LB*DEttlVill 1**1* 4CAS1K
rCTS4r4L|T4HJi (4*1*1
IT 4aCEIT*Tti
a tutu 1 am*
tlOIA
"Infinity is the abyss where
our thoughts get lost."
BAILLY Jean Sylvain
(1736 -1793)
Astronomer, politician.
Bailly became mayor of
Paris after the destruction
of the Bastille in 1789.
A royalist, Bailly was
guillotined in 1793.
Islmfimi */* U jo*/" ok it pvd**t not funt***^
B4ix.ur Kid. J* i'
TUBING*
*fff» jOiL QEOaQ COTTAM.
1 ? 9 5
L'HUILIER Simone
(1750 -1840)
The Swiss mathematician Simone L'Huilier made essential
contributions to the theory of limit values. The notation "lim" was first used in
print by L'Huilier.

781
23.1 A Retrospect
p. 361
p. 362
p. 360
Limit values have previously been discussed primarily in
Chapters 8, 10, 13, and 22; values in the form of sums,
differences, products, quotients, or power expressions of zero and
infinity require special attention.
Determinate Forms
oo — oo = — oo
oo + oo = oo
oo • oo = oo
n,—
o°°, V^
Undefined Forms
a
— where a is a non-zero real number, or <*>
Indeterminate Forms
oo — oo
0 -oo
0 oo
0 ' "
0°, oo0? 1~
cTht Limit of 9^ptfdryness
"Will you please play with me" he as fed.
"Certainly not," said the lamb. "In the first place, I cannot get into
your pen, as I am not old enough to jump over the fence. In the second
place, I am not interested in pigs. Tigs mean less than nothing to me.}}
(((What do you mean, less than nothing?" replied Wilbur. "I don't
thinly there is any such thing as less than nothing. 0\[othing is absolutely
the limit of nothingness. It's the lowest you can go. It's the end of the
line. 9iow can anything be less than nothing? If there were something
that was less than nothing, it would be something - even thongh it's just
a very little bit of something. *But if nothing is nothing, then nothing
has nothing that is less than it is."
"Oh, be quiet!" said the lamb. "Qo play by yourself I I don't play
with pigs."
E. B. White, Charlotte's Web (1952). Illustrated by Garth Williams.

782
Chapter 23 INDETERMINATE LIMITS
23.2 L'Hospital's Rule
BERNOULLI Johann
(1623-1708)
de L'HOSPITAL Guillaume
(1661-1704) Francois
Many limits of an indeterminate form can be evaluated by
a formula that is probably due to Johann Bernoulli but goes
by the name of L'Hospital's rule; L'Hospital was a pupil of
Bernoulli's.
L'Hospital's rule states that:
Iff and g are differentiate functions on an open interval
that contains a, and
or
then
1 i m f (x) - 0 ;
x —> a
1 i m f{x) = ± oo
x —> a
f (x)
lim —rr = lim
lim g(x) = 0
x -^ a
1 i m g (x) = ± oo f
x —> a
f'M
provided that gx (x) ^ 0, if x * a .
Thus, to apply L'Hospital's rule we differentiate the
numerator and the denominator separately.
After differentiation, the quotient sometimes still appears in
indeterminate form. The rule is then applied - in a repetitive
manner - to the quotients of the derivatives.
Even if the quotients of the derivatives continue to yield
indeterminate forms, the original quotient may have a limit; it
must then be determined by other methods.
One must remember that L'Hospital's rule applies only for
indeterminate forms.
Proving I/Hospital's Rule
L'Hospital's rule can be proved by using the generalized
mean-value theorem. In this text, we base the proof on
expansion into Taylor's series, assuming that f and g have such
0 oo
expansions. We distinguish two cases, tt and — , which are
treated separately.
Case 0/0
Consider
lim
x —> a
fix)
f(a) = 0 ; g(a) = 0 ,
and expand the functions f and g in accordance with Taylor's
series. Since f(a) = g(a) = 0, the first term of both series
will be 0:

Section 23.2 L'Hospital's Rule
783
f(x) = 0 + (x-a)f'(a) + j (x-a)2f"(a) + ...
g{x) = 0 + (x-a)g' (a) + j (x-a)2 g" (a) + ...
Thus, i
f (x) j\x-a)f* (a) + -^ (x ~ a) f " (a) + •••
^x) " (x-a)g'(a) + j (x-a)2 g" (a) + ... !
which may be rewritten
„ . (x-a) I/*' (a) + j (x -a)f " (a) + ...J
g(x) (x-a) [g'(a)+ { (*-a)£"(a) +
and, after cancellation,
f (x) f* ^ + 7 (x ~ a^f " (fl) + •••
^(x> ~s'(a) + j (x-a)g"(a) + ...
When x tends to a, this becomes
f(x) f'(a) + j(x-a)fn(a) +.
lim tt = lim
]
x^>aSM x^a g'(a) + j (x-a) g" (a) + ...
f'(a) + j (a-a)f"{a) + ...
g' (a) + j (a -a) g" (a) + ...
/'(a) ,. fix) ..,,. n
Iff (a) = #' (a) = 0, this formula develops into
,. fix) .. J (x-a)f"(a) + ... f»(x)
lim , . = lim -: = lim
g(x)
x —> a
and so on.
*->a o- (jc-a)#"(a) + ...
a
*"(*)
Case Woo
Consider
lim
x —> a
f(x)
g (x)
lim f(x) = ±°°; lim g(x) = ± °° .
x —» a
x —> a
If the limit lim
x —> a
f\x)
g'(x)
exists and f% and gy are defined in an
interval about a, though not necessarily at a, and g\x) * 0
in some interval about a (though not necessarily at a), then
v f^ v /"(*) w ^ ^
lim —7-t- = 11 m , , v . We find
g (x) g (x)
x—>a& v 7 * —^« e v 7
a
T ^^) 1- ^ (*)
lim —7-t = lim —:—
a
g (x)
(1)
x —»a
/"(*)

784 Chapter 23 INDETERMINATE LIMITS
and, by the case 0/0,
lim —-— = lim
x —> a ——__ x —> a
fW [fix)
By the technique of differentiation,
1 V g'(x)
,. J? <*>J .. Iff (*)]2 1 /,. /(*)
lim = lim ,, . , , = ^,,, -lim
2
x —> a / -1- \ £ —> a
f fa) ,. /"fa) " „ *(*)
fix)) |/(x)]2 ^a * fa)
and returning to (1), we now have
.. fix) 1 (,. f(x)\2
lim —:—r = * * , \ ' lim —TT
# (*) ,. / (jc) g (x)
and, finally,
v /'(*) .. fix)
lim , , . = lim
x —> a
£'(*) *->,,* <*>
Applying L'Hospital's Rule
Indeterminate Quotients
-. 1 - cos x . sin x _. cos jc 1
• lim = lim = lim =— .
x^>0 X x _> o 2 JC * ^ 0 2
e* — e ~ * e* + e ~ * 2
• lim—: = lim = — = 2.
x_>0 sin* x-^o cos* 1
In jc 1/jc
• lim —— = lim —— = 0.
p* p*
X —><» X —>oo
e2* 2 • e2* 4-e2* 8-e2*
• lim —— - lim — = lim — = lim —-— = ©o
r3 Qr2 6 jc 6
£ _>oo ^ £ —>oo *■* -A* X —>©o JC —>oo
Indeterminate Products (0 • «>)
V /1 • \ + T X " Sil1 X
lim (1 - sin x) • tan jc = lim "TIT
*->w/2 *->w/2 1/t;anJC
i • — COS X . . 2 |~v
= lim - - = lim suizjc • cosjc = 0
tan2 jc cos2 jc

Section 23.2 L'Hospital's Rule
Indeterminate Differences (°° — °°)
,• f1 1 ^ ,. In (!+»)-«
urn — -:—7T r = lim -—— —
x^0\x ln(l+*); ,_,„*• ln(l+*)
,. 1+x .. - x
= lim = lim
785
1+x
_v -1 -1 1
" J^n . 1+* , ,„ "1 + 1 + 0" 2 "
*->0 1+- + In (1 +x)
1+x
Indeterminate Powers (0°; °°°)
lim xx = lim ex' ln * =
x—> 0 + x —»0 +
In x
lim x • ln x - lim .. .
x->0 + x->0 + ,X
= lim (-x)
x—> 0 +
Thus,
lim #* = e° = 1.
x —> 0 +
/ lim x
e^x->0 +
= lim
*—>() +
= 0.
ln
*);
1/*
- I/*2
lim x^x = lim e(1/lnx)lnx = e1 = e.
x —> 0 + x —> 0 +
lim V* = lim e^1/x)]nx = ef lim — ln x |;
,. In* 1/x 0
lim = lim —— = -=0.
_^ x v 1 1
X —> oo X —> °°
Thus,
lim yx = e° = 1.
X —> oo

786 Chapter 23 INDETERMINATE LIMITS
The concept of limits is often hard to comprehend. However,
even an illustration may be enigmatic.
Are the three rings illustrated above joined or not?
Though none of the rings goes through any of the others, they
are assembled in such a way that if one is removed, the
remaining two fall apart.
We may conclude that taken three at a time the rings are
joined, but in pairs they are not.
These rings are part of the coat of arms of the noble family
Borromean Rings Borromeo-Arese of Milan, the source of the name Borromean
rings. The Borromean family's most illustrious members are
Cardinal Carlo Borromeo, canonized in 1610, and Cardinal
Federico Borromeo, founder of the Ambrosian Art Gallery in
Milan. Their shield bears the inscription Humilitas:
Borromeo-Arese
- yAilano -

787
Chapter
24
COMPLEX NUMBERS REVISITED
24.1
24.2
24.3
24.4
24.5
24.6
Introduction
Sums and Differences
Products and Quotients
Powers
Roots
Logarithms
Page
788
789
790
791
793
794

788
Chapter 2A COMPLEX NUMBERS REVISITED
24.1 Introduction
Chapter 3
Chapter 4
Chapter 9
Chapter 17
p. 87
p. 131
pp. 295 et seq.
p. 633
p. 87
Cartesian Form of
Complex Numbers
We introduced the imaginary unit and complex numbers in
Chapter 3 and described fundamental principles for their
handling in Chapter 4; in Chapter 9, we discussed complex
roots (solutions) of equations, and in Chapter 17, fractal
behavior in the complex number plane. In this chapter we
shall explore the use of vectors for calculations with complex
numbers, and learn to find their roots and logarithms.
If every point in an orthogonal number plane corresponds to a
specific complex number, the plane is called a complex
number plane.
Points on the x-axis represent real numbers, and points on the
y-axis, imaginary numbers; the coordinate axes are the real
axis and the imaginary axis.
In Cartesian coordinates, a complex number
is shown thus:
z-x+yl
Imaginary axis
A
y
X
Real axis
Polar Form of
Complex Numbers
Argument
p. 789 In a plane polar coordinate system, the complex number z
is represented by a radial vector, whose terminal point is
z = x + y\ and whose angle 6 with the positive real axis is called
its argument; the argument is determined only up to 2n rad, or
multiples of 2n rad.
Im(aginary axis)
(*, yi)
arg = 6
Re(al axis)
Modulus
Radius Vector
The length of the vector representing the complex number z is
denoted r; thus,
r = Vx2 + yi
where r is the modulus or radius vector (or absolute number) of
a complex number.
For a complex number z, we have
r = \z\ = \x+yi\ = ^Jx2 + y2 .

Section 24.2 Sums and Differences
789
Consider a right triangle whose hypotenuse is r. If 0 is the
acute angle opposite to side y and adjacent to side x, then
which gives
x v
cos 0 = — : sin 0=-,
r r
x = r cos 0; y = r sin 0
Polar Form of
Complex Numbers
Even if 0 is not acute, a complex number z = x + y i in Cartesian
form has the polar form
z = r cos 0 + (r sin 0) i
or
z = r ( cos 0 + i sin 0).
Find the modulus, argument, and polar form of z = V3 - i.
The modulus is
\z\ = |V3 + (-i)| = V(V3)2+ (-1)2 = 2.
To find the argument of a complex number, it is necessary
first to locate the proper quadrant in the complex plane; going
clockwise, the angle 0is negative.
1
Re tan 0 =
V3'
The argument 0 = arctan
r
v
V3
= -k/6 rador-30°.
The polar form of z is z - r ( cos 0+i sin 0);
z = 2 [cos (-k/6) + i sin (-n/6)] .
24.2 Sums and Differences
Addition and subtraction of complex numbers correspond to
the addition and subtraction of vectors.
Let vectors a and b represent the complex numbers 2 + 5i and
p. 604 6 - 3i, respectively; then, by the parallelogram law of addition,
we obtain the sum = (8 + 2i) and the difference = (-4 + 8i):
II
i i"(-4, 8d ~!
< i i ■ < i
i„ / ; i-\tfi ....:
i /i ! \* i
/--[""] [-¾ "■"■
L*i-j ! \\___
n
k , , , ,
NO(2i5i) i 1 j j 1
■*--/-•; ;.-^%j^-—< « J J i
~f*\ i rxlH^sri
1 : i+Oai—^^7
_.# 1 1 dQL-^**?--* J J.-J.J -1
L i_ 1 LYiA^bJr.fcz oi\\
; u : ; \07 -ply;
i i i i i i i i i
■ i i i i i i i i
Re

Chapter 24 COMPLEX NUMBERS REVISITED
24.3 Products and Quotients
To provide a geometric interpretation of the product of two
complex numbers, we consider their expressions in polar
coordinates,
z = r (cos 6 + i sin 6); w = s (cos (/> + i sin ¢).
We obtain
zw = r s (cos 6 + i sin 0) (cos ¢+1 sin 0)
= r s (cos 0 cos 0 + i cos 6 sin 0 + i sin 6 cos 0 + i2 sin 0 sin ¢)
= r s [(cos 6 cos 0 - sin 6 sin 0) + i (sin 6 cos 0 + cos 6 sin 0 )] ,
and, finally,
p. 509 zw = rs [cos (0 + 0) + i sin (0+0)] .
Compare the preceding product formula with the polar forms of
the complex-number factors
z = r (cos 6 + i sin 6); w = s (cos 0 + i sin 0)
modulus (zw;) = | z m; | = r s
arg(z w) = 6 + ¢,
where 6 + <f) are determined only up to 2n rad, or multiples of
2n rad.
The quotient of the same two complex numbers is
z r (cos 6 + i sin 0)
w ~ s (cos 0 + i sin ¢)
r (cos 0 + i sin 6) (cos 0 - i sin ¢)
s (cos 0 + i sin 0) (cos ¢-1 sin 0)
r (cos 6 cos 0 - i cos 0 sin 0 + i sin 0 cos 0 - i2 sin 0 sin ¢)
s (cos20 - i cos ¢ sin 0 + i sin ¢ cos 0 - i2 sin2 0)
_ r (cos A cos ft + s*n A s*n ft) + i ( sin A CQS 0 - CQS A sin 0 )
s (cos2 0 + sin2 ¢)
and, finally,
508'509 z r r ^ ^ • • //i am
— = — cos (0-0) + 1 sin (0 - 0)
W1^n modulus
z
w
r
s
arg
fz ■
= 0-¢.
yW '
Find the product z w and the quotient z /w of the complex
numbers
z = 2 ( cos 30° + i sin 30°); w = 3 ( cos 170° + i sin 170°).
\zw\ =2x3 = 6. \z/w\ = — .
arg (zw ) = 30° + 170° arS (*'">) = 30° " 170°
= 200°. =-140°, or 220°.
z h; = 6 (cos 200° + i sin 200°) — = f (cos 220° + i sin 220°)
* -5.638-2.052 i. * -0.511 - 0.429 i .

791
24.4 Powers
de Moivre's Formula
deMOIVRE Abraham
(1667-1754)
We have established that the product of two complex numbers
z = r (cos 0 + i sin 6); w = s (cos <j> + i sin ¢)
i s
z w = rs [cos (0 + ¢) + i sin (0 + ¢)] .
With w = z and (/> = 0, we may write
22 = r2 (cos 0 + i sin 0)2 = r2 (cos 2 0 + i sin 2 0)
and, similarly,
z3 _ r3 (cog £ + i gm ^)3 _ ^3 (cos 3 q + j sm 3 0)
Extending to ra factors, we obtain de Moivre's formula, or
theorem,
zn _ r/i (cos ^ 0 + i sin n 0)?
which holds good for all rational n. The formula is named for
the French mathematician Abraham de Moivre.
Find the value of z5 when z = -1+ i\3.
\z\ = -\/(-l)2 + (V3)2 = 2;
V~3 „ 2tc .
0 = arctan —— ; 0= -r— rad
Thus.
•-Re
Consequently,
z = 2
^ 2tc .
2 7t
V
cos -r—+ l sin —
z5 = 25
and, according to de Moivre
f 2n . . 2 n
\
cos ~^~+ i sin —
z5 = 32
f 10 7C
10 7C
cos
+ l sin
= 32
f 4 7C
V
4 7t
V
cos -t— + l sin —
= 32
cos
u
77 + 7C l+i sin
V3
fa
^
= 32
and, finally
7C
7C
V
- cos — - i sin — I = 32
( 1
V 2
.V*
:5 =-16(l + iV3)

792 Chapter 24 COMPLEX NUMBERS REVISITED
e1X
Expanding e ** into a power series, we have
cf. pp. 774 -75 Why? f (x) = eix => f(0) =1 (e1*0^!)
fix)
f (*)
f " (x)
f '" (x)
f(4) (x)
f(5) (x)
eix = l + ix —
— eix
— j elx
— -I & ex 1X ^ __ piA
= — i elx
^ __ -I & ex 1 X — ex 1 X
— j qIX
X2
=>
=>
=>
=>
=>
=>
and, after substitutions in
ix3 x4
3 ! + 4 !
f(0)
/"(0)
/"•(0)
/""(0)
/•(4) (0)
/-(5)(0)
ix5
■+5!
= 1
«
= 1
= -1
•
= 1
•
= 1
^1 • • • •
(i*=-l)
f(x) = eix = f(0) + f'(0)x+ f 2\0) x2 + f3 ,(0) x? + ...,
we obtain
/*(x) = el:x:=l+ix- 777 - -777- + -77 +
2 ! 3 ! 4 ! 5 !
Euler's Formula
• •
elx = cosx + 1 sinx ,
where i is the imaginary unit.
Why? Since
•A/ X *A/ *A/ X *A/ *A/ X «A/ *A/ X *A/
e1* = 1 +ix- rrr - -r-r + tt + -=~r - ttt - -z~r +
2 ! 3 ! 4 ! 5 ! 6 ! 7 ! 8 ! 9 ! '"
w*Zt A*^* >v*0 >v*0 ( AptJ /v**^ 'V* ' /v*l/
- *A/ *A/ *A/ *A/ tt I «A/ *A/ *A/ *A/
= 27 + 4T~6T + 8T+***+1r ~ 37 + B7 ~ 77 + <T7
and
/y*£ /y*TT /y*0 /y*^
•A/ *A/ *A/ *A/
p. 773 cosx = 1- 27 + 47-57 + 37 + •••;
, *A/ *A/ *A/ *A/
p. 771 sinx = x - 37+57-77+97--^
we obtain
e i x _ cos x + i sin x
and, as cos n = - 1 and sin ft = 0, the special case
ein = -1 .

Section 24.5 Roots
793
i1
What one at first might imagine as "the most imaginary
number possible" is, after all, a real number; thus:
o The imaginary unit, i, raised to the power i is a real
number.
To confirm the thesis, use the special case of Euler's formula,
-1 = ei7t,
from which we infer that
i = ein/2 ;
so
ii = (ei7c/2)\
and we find that
ji = e-*/2,
which is a real number.
24.5 Roots
Let the n-th roots of a complex number
w - r (cos 6 + i sin 6); - n < 6 < n
be zi, Z2> z3 ••• zn-> aU of which satisfy the equation
zpn = w .
With de Moivre, write
Zl = wl/n _ rl/n
= r1/n (cos 6+ i sin 0)^n = r1/n ( cos — + i sin — 1
V n n /
Adding multiples of 2n to 6, which leaves w unchanged, we
find, for/? = 2, 3 ... n, that
o all n values have the same modulus (r1/n), which is the
radius of a circle whose center is the origin of the coordinate
system;
o the roots will be equally spaced around this circle, that
is, the arguments of the roots will differ by steps of 2nln
radians.
Thus, for the complex number w,
w = r (cos 0 + i sin 6),
de Moivre's theorem gives
~ _ r\ln
tp — r
6+2{p-l)n . . 6+2(p-l)n
cos + l sin
n n

Chapter 24 COMPLEX NUMBERS REVISITED
Im
A
0= K
Re
Find all cubic roots of-8
With - 8 in complex form, - 8 + 0 i,
the modulus is \ (- 8 )2 + 02 = 8 ; the argument (6) is rc,
and the roots z\, z^, z% will be
z\
8173
r
f
K
n\
cos — + l sin —
v 6 6;
*2
cos — + l sin —
V 6 6J
f n+ 2 n . . 7C+ 2 n\
=l+iV3
cos
+ l sin
V
2 (cos n + i sin n)
z3=2
f n + 2 x 2 n
cos
V
+ l sin
J
n+ 2 x 2 n\
J
( 5 n . . 5 n\
cos -5— + 1 sin -r—
V d 6 J
=l-iV3
Hence, the three cubic roots of-8 are
-2; l±iV3,
equally spaced around the circumference of a circle whose
center is the origin of the coordinate system:
Im
i
(l,iW)
^
-8 (-2,0)
Re
(1,-W3)
24.6 Logarithms
p. 793
Since i = e17l/2,we have that
K
principal In of i is — • i .
A complex number z = x + i y may be written in polar form as
z - r (cos 0+i sin 6);
we obtain for the logarithm
In z - In r + In (cos 0+i sin 6).

Section 24.6 Logarithms
795
Principal Logarithm
cf.p. 788 (z = 1 + 1-¾)
Since
we have
or
cos 0+ i sin 0= el9; lne10=i0,
In z = In r + i 0,
lnz = lnr + i(6±2n n).
Thus, a complex number has infinitely many logarithms,
differing by integer multiples of 2ni. Of these, by convention,
the principal logarithm is the one whose imaginary part is
contained in the interval
-7t<Imz<+7t.
Find the principal natural logarithm of z = 1 + i .
\z\ = Vl2+12 = <2.
1 K
6 = arcsin
V2
4 *
= V2 cos -r + i sin —
In z = In V2 + i — ± 2 n n \ , where ra is any integei
The principal natural logarithm of z is
In V2 + -^ ^ 0.346 + 0.7854 i.
full of anticipation, Lisa and Stanley had brought their imaginary
childhood friends to the lecture, only to find that imaginary
numbers aren't imaginary at all, as you can imagine.

Chapter 24 COMPLEX NUMBERS REVISITED
From Georg Reisch, Margarita phylosophica (first edition, 1496; here
reproduced from a 1583 reprint).
In the above illustration, theology watches over the liberal
arts. In the Middle Ages, theocratic dogma did not support the
introduction of new concepts in learning, such as negative
numbers and complex numbers.
The appellation liberal arts - for geometry, astronomy,
arithmetic, music, grammar, dialectics (logic), and rhetoric -
dates from antiquity; liberal implies that such study was the
privilege only of free citizens.

797
Chapter
25
EXTREMA AND CRITICAL POINTS
Page
25.1 One Independent Variable 798
25.2 More than One Independent Variable 815
25.3 Functions with Restrictions 822

Chapter 25 EXTREMA AND CRITICAL POINTS
25.1 One Independent Variable
Inflection Point
When discussing methods of locating points of relative
maximum, points of relative minimum, and points of
inflection, the expressions concave upward and concave
downward are less open to misunderstanding than the
expressions concave and convex.
At the inflection point a curve changes from concave upward to
concave downward, or vice versa. The slope of the tangent
line may have any value.
concave
upward
concave
downward
concave
upward
concave
downward
concave
upward
concave
downward
concave
upward
ive\
concave
downward
Relative (or Local) Maximum
Relative (or Local) Minimum
Stationary Points
At points of relative (or local) maxima, points of relative (or
local) minima, and inflection points with horizontal tangent
lines, called stationary points, the first derivative, if it exists,
must be zero; the first derivative is undefined at points where
the tangent is vertical.
x
Critical Points
Any point on the curve of a function y = f(x) where the slope of
the tangent line of the curve is zero (tangent parallel to x-axis)
or undefined (tangent parallel to y-axis) is called a critical
point. In the above graph, the slope of the tangent line is

Section 25.1 One Independent Variable
799
undefined at the critical points x\ (inflection point), x<i (local
minimum), and x% (local maximum), and zero at the critical
points X4 (local minimum), x$ (local maximum), xq
(inflection point), and xq (local minimum). The slope of the tangent
line at an inflection point may have any value, as exemplified
by the inflection points at x\ and xq.
Relative Extrema and Critical Points
Relative extrema always occur at critical points of the
function:
If f (x) has a relative extremum at x = c, then the first
derivative of f(c) is either zero or undefined at c.
Although a relative extremum is always located at a critical
point, the opposite does not apply; critical points exist also at
other locations.
To distinguish between different kinds of points of interest of a
function that is twice differentiable for all values of its
domain, we refer to the graph below, which highlights points of
relative maximum, relative minimum, and inflection in a
function y = f(x) and the corresponding values of yy = f(x) and
A
fix)
fix)
f " (*)
- X
X
X

Chapter 25 EXTREMA AND CRITICAL POINTS
As illustrated in the graph on the preceding page, reproduced
here to the left in two parts, the function fix) and its first and
second derivatives have the following particulars:
1. Between A and B, fix) increases as x increases, producing
a positive slope of the tangent; the first derivative is
positive between A and B.
2. Between B and C,fix) decreases as x increases,
corresponding to a negative slope of the tangent; the first
derivative is negative between B and C.
3. At a point of relative maximum, B, the function changes
from increasing to decreasing; the first derivative
changes from positive to negative, and is zero at the point
of maximum.
Since at a relative maximum the derivative of fix)
changes from positive to negative, it follows that the
second derivative is negative at a point of relative maximum.
4. At a point of relative minimum the function changes from
decreasing to increasing; the first derivative changes
from negative to positive, and is zero at the point of
minimum.
Since at a relative minimum the derivative of fix)
changes from negative to positive, it follows that the second
derivative is positive at a point of relative minimum.
5. When the curve is concave downward - e.g., between
A and I\ - the first derivative is decreasing and,
consequently, the second derivative is negative.
6. When the curve is concave upward - e.g., between I\ and
/2 - the first derivative is increasing and, consequently,
the second derivative is positive.
7. At an inflection point the curve changes from concave
downward to concave upward ie.g., at I\) as the first
derivative changes from decreasing to increasing, or
from concave upward to concave downward (/2) as the first
derivative changes from increasing to decreasing, which
implies that the second derivative passes through zero and
changes its sign.
When the tangent line is horizontal at the inflection point
(at /5) and, consequently, the first derivative is zero, the
second derivative still changes its sign as it passes
through zero, thereby distinguishing the point from an
extremum, for which the second derivative would be either
positive (a relative minimum) or negative (a relative
maximum).

Section 25.1 One Independent Variable
801
Downward Concavity
Theorems describing downward or upward concavity are
cornerstones for the practical use of differentiation.
If fix) is a function that has a second derivative f " (x) < 0
for all x in the interval ]a, b[, the graph of fix) is concave
downward in ]a, b[.
Let x1 and x2 be any points in ]a, 6[, where x\ <x%\ according to
p. Ill the restricted mean-value theorem there will be at least one
value c for which
^ (C) = xTTx ' Xl<C<^-
! i-^ x
Since the denominator of the above function is positive, the
slope of the tangent (the first derivative) of the function is thus
decreasing in an interval where the second derivative is < 0.
Upward Concavity
Analogous to the case of downward concavity, we have for
upward concavity:
If f (x) is a function that has a second derivative f " (x) > 0
for all x in the interval ]a, 6[, the graph of f(x) is concave
upward in ]a, b[.
fix)
! L_^. x

802
Chapter 25 EXTREMA AND CRITICAL POINTS
Summary
Consider a function f{x) that is twice differentiable for all x.
Let f ' (x) = 0; solve for x; x = c.
We then have:
o
o
If f " (c) < 0, c represents a relative maximum of the
function.
If f " (c) > 0, then c represents a relative minimum of the
original function.
Examples:
f (x) = -x2+ 4x
fy(x) = -2x + 4
f'(*) = -2
Let f ' (x) = 0; solve for x.
Critical point x = 2.
Since the second derivative is
negative
the critical point is a
relative maximum.
g (x) = x2+ 4x
g ' (x) = 2x + 4
g " (*) = 2
Let g ' (x) = 0; solve for jc.
Critical point x = - 2.
positive
relative minimum.
If the first derivative is undefined or calculation of the second
derivative is cumbersome, it may be more convenient to
examine the change of sign of the first derivative at the
critical point in order to define the nature of the function:
a maximum, a minimum, an inflection point, or some
undefined irregularity.
The observations of the sign of the first derivative should be
made at points straddling the critical point, and as close to it as
possible. The following rules apply:
f (x) has a maximum a minimum an inflection
point
if f' changes + + - - + + no change
Locate possible maxima, minima, and inflection points of
y = 2 x3 - 3 x2 .
dy
dx
Let
= 6x2 — 6x
d2y
dx2
= 12jc-6.
6x2-6x = 0; x = 0, x = 1,
indicating possible extrema at x = 0 and x = 1.
Testing the second derivative for the above values of x gives
12 • 0 - 6 = - 6, implying a relative maximum at x = 0
and
12 • 1 - 6 = 6, implying a relative minimum at x = 1 .

Section 25.1 One Independent Variable
803
The corresponding values of y are
y = 0 (relative maximum)
and
y =
1 (relative minimum).
To find possible inflection points, set the second derivative
equal to 0:
12x-6 = 0; x = -
indicating a possible inflection point at x = -r .
The second derivative is negative for x < -r and positive for
x >2*. Hence, the value of the second derivative changes sign
as it passes through 0, and ( o" > - o") is an inflection point.
(0,0)
x
(1,-1)
The function y has a relative maximum at (0, 0), an inflection
point at ( ^ , - o" ), and a relative minimum at (1, -1) . •
The example below typifies how the value of a relative
minimum may exceed the value of a relative maximum.
Locate possible maxima, minima, and inflection points of
xd
y =
x + 1
dy x2 + 2 x
dx ""
d2y
dy
d x
(x + 1)2 ' dx2 (x + 1)3
= 0 gives x = 0 and x = — 2.
d2y f positive for x = 0
o ^13 i
dxz I negative for x = - 2,
implying a relative minimum at (0, 0) and a relative
maximum at (- 2, - 4).
The second derivative cannot equal 0; consequently, the curve
has no inflection point. •

804
Chapter 25 EXTREMA AND CRITICAL POINTS
Endpoint Extremum
Absolute Maximum
Absolute Minimum
When the domain of a function contains one or more closed
intervals, points on the curve corresponding to the endpoints
of the intervals may represent endpoint extrema. Absolute
maxima and absolute minima may be either endpoint
extrema or relative extrema.
Locate absolute extrema of
h (x) = jc3 - 6 jc2 + 9 x
6
domain:
[i4S
125
The endpoints are
(I'31) and (4|
6
125
■)•
Determine the first and second derivatives,
dx
= 3jc2-12jc + 9;
&2y
dx2
= 6x-12
x'
4jc + 3 = 0; (x-3)(x-l) = 0,
indicating possible extrema at x = 3 and x = 1.
d2v f positive for x = 3
9 *s i
dx [ negative for x = 1,
implying a relative minimum at (3, 0) and a relative
maximum at (1, 4).
Since 4 < 6 t^f , the endpoint at (4-,6 ToF ) is an absolute
maximum; and since 0 < 3 jr, the relative minimum at (3, 0) is
also an absolute minimum.
(4. 6 lig-)
/i(jc) = jc3 - 6jc2 + 9jc
(3,0)
2 3 4 5
JC

Section 25.1 One Independent Variable
805
Extrema at Points Where a Function Is Not Differentiable
Continuity of a function at a point does not necessarily imply
differentiability at that point and, as illustrated in the example
below, it can be expected that extrema may exist at points where
the function has no derivative.
The function f (x), depicted below, is continuous between a and
c. In spite of the fact that the function is not differentiable at x\
and X2, a relative minimum does indeed exist at x\9 and a
relative maximum exists at x%. The relative extrema here are
also absolute extrema:
x
Problems
Find the values of the relative extrema and sketch the graphs
of the functions
g (x) = 1 - x2 and G (x) = | 1 - x2 | .
Withy =g (x),
dy
dy _ dfy _
dx dx*
-;— = 0 gives x = 0, suggesting a possible extremum.
d2y ,
dx2
is negative for x = 0, implying a relative maximum of
g (x) at (0, 1) , which is also an absolute maximum.
g (x) intersects the x-axis for 1 - x2 = 0; the intersected points
are (- 1, 0) and (+ 1, 0).
g(pc)
i
g(x) = l-x2
x

806
Chapter 25 EXTREMA AND CRITICAL POINTS
1 : 1-x2 < 0
|l-*2| = x2
11-x2 | = 1-x2 ; 1-x2 > 0
G'(x) = 2x
G'(x) = -2x
G'(x) = 0 will give x = 0.
G' (jc) changes sign from positive to negative as it passes
through zero. Thus, G(x) has a relative maximum at (0, 1).
G{x) = 1- x2 meets the x-axis where
x'
0; xi = 1, X2 = — 1.
G(x)
i
G« = II-x21
JC
Since G(x) = \ 1 - x2 \ has only positive values it returns
upward at the relative minima (-1, 0) and (+ 1, 0), which are
also absolute minima.
Determine extrema and inflection points of
f(x) = -r-T
xL + 1
and sketch the graph of the function.
f (*) = "
2x
fix) = 0 =
(x2 + 1)2 :
2x
f"(x) =
6x2-2
(x2 + 1)2
(x2 + 1)3
= 0 gives x = 0.
6x2-2
Since f " (0) = — < 0, a relative maximum (0, 1), which
(xz + l)6
is also an absolute maximum, is located at the critical point.
To determine inflection points, let the second derivative equal
0, solve for x, and test for changes of signs when f " (x) passes
through 0:
6x2-2
(x2 + 1)3
= 0
x
±|V-3

Section 25.1 One Independent Variable
807
Test the second derivative for values of x in the intervals
between the critical points; for instance:
]-jV5,o[
«--!=> '"(-!)-
(- 02 -
[(-*)•♦']
< o
]0,|V3[ * = + £=> />■■(!)=_
(i)
[(I)2-]
< 0
]|V^,oo[
*=+1^r(1) = 1(1^4 >0
[(1)2+1]3
Since the second derivative changes its sign between
and between
# = -1 and x = - —
x = + — and x = + 1,
the curve has inflection points at ( -:r V3, t ) and ( + o"V3, j J
y
1
/(x) = l/(x2+l)
x
Find the dimensions (radius and height) that yield the
smallest external surface of a 1000-m3 right cylindrical container.
With the radius r m and the height h m,
1000
h = r m.
The total surface area S is a function of r,
1000
S(r) = 2nr2 + 2nr
= 27tr2 +
2000
S '(r) = 47tr-2000r-2; 5 " (r) = 47C + 4000r"3 .
4nr-2000r-2 =
/500
m.

808
Chapter 25 EXTREMA AND CRITICAL POINTS
Since S
11
m
= 4tc + 4000
J
m
500
VT.. ,
> 0, the surface
area has its minimum value when
-V¥
& 5.42 m ; h =
1000
f 3/ \2
500
# 10.84 m.
K
m
)
Find the base angle and the height yielding the maximum
volume of a right circular cone whose mantle generatrix is a.
Wja? - x* A
The volume of the cone is
V(x) = 3 7c(a2-
jc2) x = 7tk (a2 JC
2nx < 0 .
X3)
1 o o a v3
—7C (az - 3 xz) = 0 ; the height x = —-—
With the sought angle a,
aV3
h
sin a = — =
a
a
V3
3
a = 35.26
Find the largest possible volume of a right circular cone
inscribed in a sphere of radius R
Let the radius of the base of the cone be r, and its height x; the
height of the center of the sphere above the center of the base of
the cone will be (x - R):
x
,
R
1
R
^"•^^
'
R2 = r2 + (x-R)2: r2 = 2Rx-x2.

Section 25.1 One Independent Variable
809
The volume of the cone is
V (x) = — n r2 x .
— n (2 R x - x2) x = — n (2 R x2 - x3).
V ' (x) = -k(4Rx-3x2); V " (x) = -n(4R -6x).
1 4R
— n (4 R x - 3 x2) = 0 ; x\ = 0 (discarded); X2 = -r~
Since
^•■'¥-1-
4i? -6
4i?
= -7c(-4i?) < 0,
4R
x =
represents a relative (and absolute) maximum.
^max — o ^
2R
4R
4R
32nR3
81
Find the smallest possible volume of a right circular cone
circumscribed to a sphere of radius R.
Let the radius of the base of the cone be r, and let the distance
between the center of the sphere and the vertex of the cone be h.
The distance between the vertex and the point of contact
between the cone and the sphere is ^h2 - R2 .
The volume of the cone is
V(h) =\nr2{h + R).
Similarity of triangles ABC and DOC gives
r h + R V/i + R^h
R ^h2-R2 V(/i + R){h-R)
R

810
Chapter 25 EXTREMA AND CRITICAL POINTS
Substituting for r2 inV (h) gives us
V(h) = \nR2j^ (h + R)
kR2 (h + R)2
3 h-R
For convenience, let u =
nR2
Vl(h) = u
and
2 (h2-R2)- (h+R)2
(h-R)2
V"(h) = u
(h-R)2(4h-2h-2R)-[2 (h2 - R2) - (h + R)2] 2 (h-R)
(h-R)4
V'(h) = 0 yields
hi = 3R ; h2 = -R (discarded).
Since V " (3 R) > 0, h = 3 R represents a relative (and absolute)
minimum,
nR2 (3R + R)2 8kR3
V,
min
3 SR -R
What is the largest rectangle that can be inscribed in the
ellipse
x2 y2
— + — = 19
9 4
Let the corners of the rectangle be
±*o; ±yo
and find the maximum value of the area A = 4 xq y$ .
(-*o> yo)
(~x0> -^o)
(*o, yo)
x
(*o> -yo)
Solving for y yields
y
^ I 4x^~
hetf(x) = xy\ then
x) = x*\J
f(x) =
4x2
9 *

Section 25.1 One Independent Variable
811
fix) = * (1)(4-
4xA-mf 8
4*2
4
9A 4
4x2
4-
4 x'
Letting f' (x) = 0, and solving for x,
x0 =
3^2
and y0 =
f
4-
3^2
^ 2
2^ \ 9
The area of the rectangle is
A = 4 • xoyo - 12 area units.
.-&
(*M
X
(i£.-vs)
p. 511
Find all extrema of
fix)
in the interval ]0, 2 tc[ .
= 2 sin x + sin 2 x
f ' (x) = 2 cos x + 2 cos 2 x
= 2 (cos x + 2 cos2 x - 1)
= 2 (2 cos x - 1) (cos x + 1)
/*' ' (x) = 2 [(2 cos x - 1) (- sin x) + ( cos x + 1) (- 2 sin x)]
Let/*'(:*;) = 0; solve for x:
2 cos #-1 = 0
1
cos x + 1 = 0
cos x - -1
JC = 71
COSX =
71 571
X = 3 ;X =~3~
Thus, the function has the critical points
71 571
X — ~, X — 7l, X — _

Chapter 25 EXTREMA AND CRITICAL POINTS
Examining the second derivative at the critical numbers, we
have
= 21 2 cos^-+ 1 II- si
and
5 tc V . 5 n
cos "q"+ 1 II - sin
n\ ( tc
n3"J + tC°S3
TC
2 sin — ] < 0 (rel. max.)
+ cos
5 tc
2 sin
5 tc
> 0 (rel. min.).
On the other hand, we find
/"'(tc) = 2 [(2 costc + 1)(- sin tc) + (cos tc - 1) (-2 sin tc)] = 0,
giving no information about a local extremum at x = 0, but
suggesting the possibility of an inflection point.
In the interval J tt , -r— [
and
/"" < 0 for x < tc
f">0 for x > tc
implying that x - tc represents an inflection point.
Hence,
==^ T max :
TC
at x == ^r
TC IK
2 sin —+ sin 2 • — I & 2.6
atx = tc
5tc
atx = —r-
finfi - 2 sin tc + sin (2 tc) = 0
/min = 2 sin~+ sin I 2 ' "3" ' **
= 2 sin x + sin 2 x
x
2.6.
» Are the inequalities ex-x > 1 and ex > ex true for all realx?
1. If e* - x > 1, then ex-x - 1 > 0 .
hetf(x) = ex-x - 1 .
Differentiation gives
f'(x) = ex-l and f"(x)
Let/"(x) = 0,
e*-l = 0; x = fl (e° =
/"'(0) = e° = 1.
= e*
l).

Section 25.1 One Independent Variable
813
Since the first derivative is 0 and the second derivative is
positive at x = 0, f (x) = ex - x — 1 must have a relative
minimum at that point, which is also an absolute minimum
on ] — oo , oo [
Thus, the inequality e* - x — 1 > 0 must be true and,
consequently, ex -x > 1 for all x.
. If e* > e x, then e* - e x > 0.
Let g (x) = e* - e x .
Differentiation gives
g ' (x) = e* - e and g " (x) = e* .
Let g'(x) = 0,
e* - e = 0 ; jc = 1 (e1 = e).
S"(l) = e.
Since the first derivative is 1 and the second derivative is
positive at x = 1, g (x) - e* - e x must have a relative minimum
at that point, which is also an absolute minimum.
Thus, the inequality ex - e x > 0 must be true and,
consequently, ex >e x for all x.
A company has 2000 units of a given product in stock and can
produce 100 more every day.
An order is received for as many units as possible. For
delivery on the date of receiving the order, the company will
make a profit of $6 per unit; every consecutive day sees a
decline in profit of $0.20 per item.
How many days of manufacturing should be used for adding
to what already is in stock to make the profit the greatest
possible?
No units were manufactured on the day the order was
received. Delivery will be made after manufacturing hours
on the delivery date. There are no other obligations of delivery
interfering with the one in question.
Let the number of days of manufacturing be x.
Express the profit as a function P (x),
P (x) = (6 - 0.20 x) (2000 + 100 x)
= - 20 Jt2 + 200 x + 12 000 .
P'Ge) = -40*+ 200.
Pn(x) = -40.
Let P ' (x) = 0,
-40*+ 200 = 0; x = 5.
As the second derivative is negative, x - 5 must represent a
maximum.
5 days of manufacturing should be added.

Chapter 25 EXTREMA AND CRITICAL POINTS
A power plant shall supply electric power to a factory situated
1000 m downstream on the opposite bank of a 200-m-wide river.
A cable is to be laid for the purpose. The cost of cable and
installation is $50 per meter on land, and $80 per meter of
riverbed. What is the most economical course for the cable?
Let the distance on land be (1000 - x) m; the distance under the
river will be
d = Vx2 + 2002 .
1000-x
x m
d
200 m
r
*****<*i*v*+**
The cost is C (x):
C (x) = 50 (1000 - x) + 80 Vx2 + 2002 .
C ' (x) = - 50 + 80
T2*
Vx2 + 2002
C ' (x) = 0
5 • V*2 + 2002 = 8x ; x * 160.1.
1000-x ^ 839.9.
d * Vx2 + 2002 = 256.2 .
With the distance 840 m on land, the distance under the river
will be 256 m.
* * A-
Had he not overlooked the variable of the nearby swamp, his
calculations for a maximum of success would have been correct.

815
25.2 More than One Independent Variable
Extrema and other critical points of functions of more than one
p. 710 independent variable are traced and identified by partial
differentiation; such functions are differentiated with respect to
all of the independent variables - one at a time, while the
others remain unchanged.
Functions of one independent variable, y = fix), can be
depicted in a two-dimensional coordinate system, whereas
functions of two independent variables, z =f(x,y), require a
three-dimensional system. To display functions of three or
more independent variables, colors may be used, although
there are no recognized rules for the use of colors in graphic
representation.

816 Chapter 25 EXTREMA AND CRITICAL POINTS
A differentiable function of one independent variable has a
maximum or a minimum where its first derivative is zero;
similarly, differentiable functions of two independent
variables will have a critical point of some description where both
their first-order partial derivatives are zero.
This critical feature may be a local or absolute maximum, or
Saddle Point minimum, with a horizontal tangent plane, or a saddle point,
which is neither a maximum nor a minimum - or,
alternatively, it may be considered both, that is, a maximum
in some directions and a minimum in other directions, as
illustrated by the drawing of a hyperbolic paraboloid on the
previous page.
The prime requisite for the existence of an extremum or a
saddle point at a point (a, b) is that
df(a,b) _ Q. y (a, b) _ Q
dx ' dy
The actual nature of the feature is decided by the values of the
simple and mixed second-order partial derivatives
d2f(a,b) d2f(a, b) d2f(a, 6)
dx2 ' dy2 ' dxdy
and of the discriminant
d2f (a, b) d2f (a, 6)
g(a, b) =
dx2 dy2 L dxdy J
d2f(a,b)
If g (a, 6) > 0, in which case the simple partial derivatives
must have the same sign, the critical point is
- a maximum
if32^6)<Q. ^ 6) < Q •
3*2 ' 3^2 <U'
- a minimum
., aV (o, 6) n 32/(q, ft) n
lf ^2— >°; -#r- >°-
If # (a, 6) < 0, the critical point is
- a saddle point.
If g (a, 6) = 0, the critical point may or may not be one of the
above types.
Find extrema and saddle points of
f(x,y) = x3 +ys + Sxy .
We have
j- = Sx2 + Sy; ^- = 3y2 + 3x.
Let both derivatives be equal to 0 and solve the system of
equations
Sx2 + Sy = 0
Sy2 + Sx = 0,-

Section 25.2 More than One Independent Variable
817
There are two critical points, (0, 0) and (—1 ,— 1).
The values x = 0, y = 0 and x = — 1, y =
/(0,0) =0
/•(-1,-1) = 1
The second derivatives are
1 yield, respectively,
32/
dx2
= 6x
32/
dy2
= 6y;
32/
dx dy
= 3
For the critical point (0, 0, 0), we find
32/(0, 0) 32/(0, 0)
dx2
dy'
32/(0, 0)
_ dx dy .
= 0 0-32 < 0.
which indicates a saddle point.
And for (-1 ,- 1, + 1),
32/(-l,-l) 32/(-l,-l)
dx2
3v2
32/(-1,-1)
dx dy
= (-6)-(-6)-32 > 0,
which indicates a relative extremum, whose nature is
determined by the sign of the second partial derivatives of f with
respect to x and y. Since
ay (-i,-i) r n &f (- i,-i) . n
= -6 < 0 ; = -6 < 0,
dx1 dyz
f (- 1, - 1) represents a relative maximum.
Thus, /"has a relative maximum at (-1, -1, +1) and a saddle
point at (0, 0, 0).

818
Chapter 25 EXTREMA AND CRITICAL POINTS
Identify relative extrema and saddle points of the hyperbolic
paraboloid
f(x,y) = x2-y2 .
df
df
dx ~2x' dy ~ ~2y-
Equate both derivatives with 0 and solve the equations:
x = 0
y = o
f has only one critical point, (0, 0, 0).
For x = 0 and y = 0, we find
/(0,0) = 03-03 = 0.
The second-order partial derivatives are
d2f
dx2
d2f(0, 0) d2f(0, 0)
= 2;
d2f
dy2
dx2
dy'
32/(0, 0)
. dx dy .
= -2;
2
d2f
dx dy
= 0.
= 2(-2)-02 < 0,
indicating a saddle point.
Thus, f has a saddle point at (0, 0, 0) but no relative extrema.

Section 25.2 More than One Independent Variable
819
Find extrema of
f(x,y) = sin 3 x cos 2 y .
The maximum and minimum values of sin3x and cos2y are
1 and - 1.
Consequently, f has
- maxima for
- minima for
sin3#
cos2y
sin 3 x
cos2y
= 1
= 1
= -1
= 1
and
and
sin 3 * = - 1
cos 2 y = - 1
sin 3 # = 1
cos 2 y = - 1
sin 3 x = 1
= -1
sin Sx
cos2y = 1
cos 2 y = - '.
K
3x = — +n2n; x =
Sx = —t- +n2n; x =
2y = 0 + m2n; y =
n n2n
6+-
n
2 +
m n
3
n2n
n
2y = n + m2n; y=—+mn.
Thus, /"has iterative maxima at
'jc_ n 2 n
6+ 3
, m n , + 1
/7C 71 2 7U K
and 1-+ —-—, - +m 7C , + 1
and iterative minima at
n n 2 n n
6 +
_ /7C 71 2 K
and I — + —-—, m n , - 1
All extrema are absolute extrema.
*t
*><P
& **<*•

820
Chapter 25 EXTREMA AND CRITICAL POINTS
Find the critical points of the paraboloid of revolution
f(x,y) = x2 +y2 .
g=2*; J£= 2y.
Equate the derivatives with 0 and solve the equations,
x = 0
y = o
Thus, /has only one critical point, (0, 0, 0).
The second-order partial derivatives are
d2f(0, 0)"
. dx2 J
d2f
dx2
d2f(0, 0)
dy2
= 2
d2f
dy2
= 2;
d2f
dx dy
= 0.
d2f(0, 0)
. 3* dy .
= 2(2)-02 > 0
indicating that f has a relative extremum.
Since 2f£i!L2!= 2 > o ; f(°' 0) = 2 > 0, the point (0, 0, 0) is an
dxz dyz
absolute minimum, the vertex of the paraboloid.

Section 25.2 More than One Independent Variable
821
Find relative extrema and saddle points of
g(x,y) = x2-y2-4x-2y + S .
Equate both derivatives with 0; solve for x and y,
x = 2
y =-1
Thus, there is one critical point, (2 , - 1), where
£(2,-1) = 0.
The partial second derivatives are
d*g
= 2
3¾
= -2 ;
d2g
dx2 ~ ' dy2 ~ ' dx dy
We have, for the critical point (2, -1),
= 0.
3¾ (2,-1) 3¾ (2,-1)
d2g (2,-1)
dx dy
= (2) (- 2) - 02 < 0
dx2 dy2
indicating a saddle point.
Thus, g has a saddle point at (2, -1, 0), but no relative extrema.

822
Chapter 25 EXTREMA AND CRITICAL POINTS
25.3 Functions with Restrictions
Side Condition
Constraint
Extreme values of functions of more than one independent
variable can be found only in the case of a side condition, or
constraint, which relates the variables to each other.
Let us study the function
z = f(x,y)= 7\l--x--y
represented by a plane in space intercepting the coordinate
axes at
x = 4; y = 6; z = 7,
the variables x and y being bound by the constraint
g(x,y): (x-2)2 + (y-3)2-l = 0,
which is a circle of radius 1 centered at (2, 3) in the x, y-plane
and a right circular cylinder with the circle as base.
The set of points (x, y, z) that satisfies both the equation z = f(x,
y) of the inclined plane and the equation g (x, y) of the upright
circular cylinder, is the elliptic section of plane and cylinder,
the upper half of which is shown in the figure. We seek the
maximum and minimum values of z on the ellipse.
z
A
z=f(x,y) = 7(l-Lx_Ly)
y
g(x, y) = (x - 2)2 + (y - 3)2 - 1 = 0

Section 25.3 Functions with Restrictions
823
Lagrange's Method of Multipliers
To find extrema of f(x, y) with the constraint g(x, y) = 0, we
introduce the function
F(x,y,X) = f(x,y) + Xg(x,y),
where X is Lagrange's multiplier, named after the French
mathematician and scientist Joseph Louis Lagrange.
We differentiate F with respect to xt to y, and to X ,
K + 2 *. V+2 te. E
dx dx ' dy dy ' 3A '
and equate them with zero to form a system of equations, which
must be satisfied at any point where a restrained extremum
occurs.
Lagrange's method of multipliers can be extended to a
function f with any number of independent variables if
supplemented by the requisite number of qualifying equations
in the same variables.
Thus, keeping within bounds, a function f(x, y, z, w) with the
constraining conditions g (jc, yt zt w) = 0 and h (jc, y, z, w) = 0
will give the function
F(x, y, z, w, X, ill) =f(x, y9 z9 w) + kg(x, y, z,w) + jli h(x, y, z, w),
with Lagrange's multipliers X and /i.
We return to the function
( 1 1 \
f(x,y) = 7\l--x--y
with the constraint
g(xyy): (x_2)2 + (y-3)2-l = 0;
find the extreme values off.
Define a new function,
F(x,y9X) = f(x,y) + Xg(x,y)
7 7
= 7 - —x - — y + X (x2 - 4 x + y2 - 6 y + 12) .
Partial differentiation of F gives
2Ajc-4A; ( ^- ] = -- + 2 Xy-6 X; \^r) = x2-±x+y2-6y+
$y
j
dx
12
Letting the first partial derivatives of F equal 0, we arrive at a
system of three equations:
T+ 2Xx-4X = 0
4
a
-+ 2Xy-6X
■ 0
x2-4x+y2-6y+12 = 0
>
J
X =
X =
8x- 16
7
12y-36
j
X ~ 2y 2

824
Chapter 25 EXTREMA AND CRITICAL POINTS
dF
Insert the obtained value of jc in -r-r- = 0,
2] -41 2^
2J+y2-6y+12 = 0
o ~ 113 „
y2-6y + rg-=0
y = 3±^-Vl3
x = 21-j-g Vl3
f(*i,yi) = 7
f(x2,y2) =7
--4^
-4^
Thus, the maximum value of /* (jc, y) subject to the constraint
(JC - 2)2 + (y - 3)2 = 1 is
/>(2-ivrsf 3-^Vm) = £Vi8
and the minimum value is
/■f2+^-Vl3, 3+||- Vl3
iK3
A
s = fl*,:y) = 7(l-i*-£y)

Section 25.3 Functions with Restrictions 825
Find the maximum and minimum values of
f(x,y,z) = 5x+y + 2z,
subject to the constraints
x + y-z -1 = 0 and y2 +z2 - 1 = 0 .
Form the function
F (x, y, z, A, /i) = 5 x + y + 2 z + X (x + y - z - 1) + /i (y2 + z2 - 1).
Equating the derivatives with zero, we have the following
system of equations, and solving for x, y, and z:
5 + *=0>> *i=l--p=r x2=1 + ^~
1 + X + 2jny = 0
2-X +2fiz = 0
x+y-z- 1 = 0
y2 + z2 -1 =0
We obtain the extreme values
V65 ^65
4 4
> yi= i= y2= -
V65 V65
7 7
zi = - -==■ z2 =
V65 ^65
and
f(*i,yi,zi) = 5 [ 1 --^=) +-^=- -2-^==- =5-V65
V657 V65 V65
/(^2,^2,^2) = 5[-p^+ II- -p^+ 2--=-= 5+V65.
W65 y V65 V65
Subject to the constraints (x+y-2:-l = 0) and (y2 + z2 - 1 = 0),
f(x,y,z) has a maximum
V V65 V65 V65V
and a minimum
V V65 V65 V65y
Find maxima and minima of
f(x,y,z,w) = xyzw
with the constraining condition
g (x9y,z, w) = Sx2 + 4y-z + Sw -84 = 0.
We introduce the function
F(x,y,z9w9X) = f(x,y,z,w) + Xg(x,y,z,w)
= xyzw + X(3x2 + 4y-z + 8w- 84)

826
Chapter 25 EXTREMA AND CRITICAL POINTS
Partial differentiation gives
rdF_
\dx
rdF_
\^dz
rdF_
KdX
= yz w + 6 Xx
= xy w - X
= xzw + 4X
=xyz + 8X
J = 3x2 + 4y-z + Sw-S4.
Equating the first derivatives with zero, we have
yzw + 6Xx = 0
xzw + 4X = 0
xy w - X = 0
xyz + 8 X = 0
3x2 + 4y-z + Sw-S4= 0
>
j
xyz w + 6 Xx2 = 0
xyz w + 4y X = 0
xyz w -z X = 0
xyzw + 8 w X = 0
y
z
w
J
3 2
= ~~ X*
2
= -6x2
_3 2
4
There are also (infinitely many) solutions where X = 0. Since
/*(jc, y, z, w) = 0 at all of them, these solutions will not affect the
final result, and are thus omitted.
dF
Insert the above solutions fory, z, and w in t-t- and equate with
zero,
3 3
3jc2 + 4--jc2 + 6jc2 + 8--jc2-84= 0; jc2 = 4, *i = 2, x2 = -2.
3 ^
yi = 2 *i2 = 6
zx = -6jq2 = -24
3 9
>•
3 9 „ "\
y-i = 2 x2z = 6
02 = 6*i2 = -24
3 2 o
>
The extreme values of fix, y, z, w) restricted hy g are
/(2,6,-24,3) = 2-6-(-24)-3 = -864
and
/(-2,6,-24,3) = -2-6-(-24)-3 = 864

827
Chapter
26
ARC LENGTH
Page
26.1 Basic Principle 829
26.2 The Catenary 831
26.3 Arc Length in Parametric Form 832
26.4 Arc Length in a Polar Coordinate System 835

Capilano Suspension Bridge, British Columbia, Canada.

829
26.1 Basic Principle
The length of an arc of a curve is approximately equal to the
sum of the lengths of the straight lines connecting subsequent
points on the curve.
Assume that f(x) is a function with a continuous derivative
over the interval [a, 6] which is divided into n intervals of
equal length with endpoints that [a, 6] = [xq, xn] :
y
(*o, yo)
a =xq x\
X2
y = fhc)
X
b = xn
p. 550
p. 716
Denoting the arc length by s, we have
s * V(*i-*o)2 + (yi-yo)2 + V(*2 -*i)2 + (y2 -yi)2 +•••
+ V(x„ -xn_{)2 + (yn -yn-i)2
or
n
i = l
If we let n tend toward infinity, then
n
s = lim X V(Axj)2 + (Ayt)2 ,
n —> oo i = l
which can be rearranged to
IV
1 +
(Ayi}
yAxt j
Axt
s = lim
n —> oo i =
Since /*(jc) is continuous on each interval [x( _ i, xj , we find by
Lagrange's mean-value theorem that there exists a value E,i
between x; _ \ and xi such that
f(Xj) - f(Xj _ i)
Since A# =yt -yt-i=f(xi) -f(xi_1) and A*; =xt -xt_lfweget
Ay,
Ajc,
=f(&),

830
Chapter 2 6 ARC LENGTH
and we may write
s= lim X Vl+tf (&Xl2 A**,
equivalent to
n —> °°
i = l
= J Vl + [/"'(x)]2 dx,
a
which is the formula for the arc length ofy = fix) over the
interval [a, b] on the x-axis.
Similarly, integration with respect to y gives
d
s = J ^1 + ig'iy)]2 &y.
Find the arc length of the curve
fix) = V^
from x = 0 to x = 4.
p. 947
dx 2
Let the sought arc length be s;
4
s =
o
v
l + l|*1/2T dx
J
o
Vi+I*
dx
Let 1 + tx = w; dr = — dw.
4 9
With the new limits
x = 0
x = 4
=> w = 1+t-0=1
4
1/ = 1 + - .4=10,
4
10
10
= J -fu | di< = | J u1/2 da = —
27
10
w
3/2
= — (V 1000 -l) = 9.073... units.
27
The presence of a radical expression in the integrand of the
formula for arc length often precludes the expression of the
indefinite integral in terms of an elementary function;
therefore, numerical integration, described in Chapter 32, is
commonly used to find the length of arcs.

831
26.2 The Catenary
p. 538
A catenary is the curve that a uniform, inextensible, and
flexible cable or wire, strung freely between two fixed points,
assumes under the influence of gravity. The catenary is an
application of the hyperbolic cosine function and has the
equation
fix) = a
fg)cl a
+ e
-x/ a
V
X
or f(x) = a
cosher
p. 702
with the vertex at (0, a) in an orthogonal coordinate system.
Let s be the arc of the catenary between x = 0 and x = b:
b
r
s =
j
0
-^1+
-\— a cosh— I dx
\d x a
b
r
j
0
V
f x
1 + sinh — 1 dx
v
Pythagorean Identity: p. 541
b
r
j
0
V
cosh — 1 dx
V a
b
r
j
0
x
cosh — dx
a
p. 725
= a sinh —.
a

832
Chapter 26 ARC LENGTH
26.3 Arc Length in Parametric Form
A two-dimensional curve may be represented by two
parametric equations, which give the coordinates x and y in terms of a
third variable, the parameter. Similarly, a three-dimensional
curve may be represented by three parametric equations,
giving the coordinates x, y, and z in terms of the parameter.
A formula for arc length in parametric form can be developed
from the formula for arc length,
u b
r ' fd^Y
a J *
a
1 +
, ■ dx .
ydxj
where s is the length of an arc of y = f(x), within the interval
[a, b] on the x-axis, where fl (x) is required to be continuous.
Consider a plane curve in parametric equations
x = x(t)
y = y(t) J
where a<t<b, and the functions x(t) and y (t) have continuous
derivatives in [a, 6].
We have
ds = ^j
1 +
Since
dy
dx
variable £; thus,
d y/dt ._ dx .,..,
-j—t-j— if -T- =£ 0, we may rewrite this in the
ds =
fdy/dtY
dx/ dt
dx
(dx/dt)2 + (dy/dt)2
(dx/dt)2
dx
(dx\2 (dy\2 dt
\
dt
-
V
dt J dx
dx
(dx\2 (dy. ,
V*J
Thus, the formula in parametric form for arc length s in the
plane is
b
r
s =
j
a
V
(dx\2
Kdt
% *
V
for a curve x = x(t), y = y(t), where a <t < 6, and the functions
x(t) andy(£) have continuous derivatives in [a, b].

Section 26.3 Arc Length in Parametric Form
833
Analogously, the formula in parametric form for arc length s
in space is
s =
V
'#; * (¾2 * m2 -
a
for a function x = x(t), y =y(t), z = z(t) where a <t <b, and the
functions x(t),y(t), z(t) have continuous derivatives in [a, b].
Find the length of the perimeter of the astroid
x = 2 cos31
y = 2 sin31
t = 71
£ = 71/2
Because of the symmetry of the astroid, the perimeter s is 4
times the length of the arc in one quadrant,
w/2
= 4 J Vt*' (*)]2+ ly' (t)]2 dt .
o
Since
x* (t) = 2-3 cos21 (- sin £) = -6 cos2 t sin £ '
/ (*) = 2-3 sin2 £ (cos t) = 6 sin2 £ cos £
we obtain
w/2
s = 4 J -y/(- 6 cos2 £ sin £ )2 + (6 sin2 £ cos £ )2 dt
0
7t/2
= 24 I V cos4 £ sin2 £ + sin4 £ cos2 £ dt
0

834
Chapter 26 ARC LENGTH
w/2
= 24 J Vsin2 t cos2 t (cos2 t + sin2 t) &t
0
7C/2
= 24 f Vsin2* cos2* d£
0
7C/2
= 12 f 2 sin £ cos t dt
0
7C/2
= 12 J sin 2£ d£
0
= -6
7t/2
0
cos 2 £ = - 6 (- 1 - 1) = 12 length units.
I baf^e apple pies in threes
for a fellow who litres to f^eep bees.
We measure arcs in parametric form
White the pies are stiCi warm!
Then we sit eating pie in the trees.

835
26.4 Arc Length in a Polar Coordinate System
To find the arc length of a curve expressed in polar
coordinates, we observe that
x = r cos 6
>
y = r sin 6 J >
as illustrated by the graph:
y
(x,y) Cartesian
(r, 0) polar
polar axis
x
Assuming that r = f (0), where /"has a continuous derivative,
we have the arc length s,
P
r
s =
j
a
V
dOJ + \d0;
d6
in which we insert
dx dr dy dr . n
-^ =-^ cos9-rsm9; ^ = ^ sin 0+ r cos 9
and obtain
s =
j
a
v
dr
-T77 cos 0 - r sin 0
d0 y
A2 /dr \2
+ 377 sin 0 + r cos 0
de
or, after simplification,
s =
j
a
V
r2 +
dQj
dO,
where s is the length of an arc of r = f{6) between 6 = a and

836
Chapter 26 ARC LENGTH
Find the length of the arc of the spiral
r = 2 e~e
between 0=0 and 6 = 2n.
0 = n
0=0 ,
2 6=2n
0=3rc/2
2tc
s =
<\](2e-°)2 +
^ [2e-qT d6
o
2tc
y(4e~2e) + Ue-2e) d0
o
2tc
2tc
= 2 J V 2e-20d0= 2^2 J e"* d0
0 0
27C
= 2^2 [-e-°] = 2^2 [-e"2* +e°]
0
= 2^2 (l -e"27C) * 2.823 length units.

837
Chapter
27
CENTROIDS
Page
27.1 Mass Point Systems 838
27.2 Plane Figures and Laminas 842
27.3 Center of Mass of Solids of Revolution 847

838
3£<*»i:>\»
27.1 Mass Point Systems
Suppose that mj, m<i ... mn are the masses of n objects attached to
a rod supported by a knife edge acting as a fulcrum.
Objects to the right of the fulcrum tend to swing the rod
clockwise, whereas those on the left move it counterclockwise.
Our task is to find a point on the rod that should be placed
directly over the fulcrum in order to obtain exact balance of the
rod and the attached objects.
To solve the problem, let the rod - when exactly balanced -
coincide with the x-axis of an orthogonal coordinate system,
and let the fulcrum be located at a point a:
Moment Arm
Moment of Mass
Centroid, Center of Mass,
Center of Gravity, Barycenter
Greek, barys, "heavy"
Let d( be the distance (x; - a), that is, the moment arm of mi;
the product of m/ and the distance di is then the moment of
mass of mj with respect to a. Assuming the mass of the rod to be
negligible in comparison with the masses, the total moment of
mass M with respect to a is the sum of all the moments of mass;
thus,
n
M = X midi •
i = \
The rod is in exact equilibrium if and only if M = 0.
If M = 0 around a point a of an object, a is the centroid of the
object, and commonly denoted ~x. Alternative terms are center
of mass, center of gravity, and barycenter.

Section 27.1 Mass Point Systems
839
In a state of equilibrium, we have
n
2^ m>i (xi - x) - 0
/ = 1
and
n
X =
X rriixi
t = l
n
X mi
i = l
If m denotes the total sum of the masses of the system,
n
™>= X mi 7
i = l
and Mq is the sum of the individual moments of mass,
n
M0= X ™>i*i,
i = l
we have
x
Mo
m
Three objects with masses 2, 3, and 6 are placed on the x-axis at
4, and x = 7, respectively; find the position of the
centroid of the system.
x =
M0 2-(-3) + 3-4 + 6-7 48
m" 2 + 3 + 6 "11
i
t 1 1 1 1 r
-3 0 4,
3 6
JC
48/11
The concept of a centroid is applicable also to systems in the
plane and in space.
Consider n objects with masses m\, m^ ... mn located in the
x,y-plane at (xi,yi), (x2,y2) ••• (xn,yn), respectively.
(*i,:xi)
y
A
0¾ y2)
x
(*n, yn)

840
Chapter 27 CENTROIDS
If My is the total moment of mass with respect to the y-axis, we
have
My = mi x\ + rri2 #2+---+ mn xn >
and similarly, if Mx is the total moment of mass with respect to
the x-axis,
Mx = m1y1 + m2y2 + ... +mnyn .
If m is the total mass of the system, the location of the centroid
(3c, y) is given by
x =
Mx
m
and
AT,
y =
m
Three masses 2, 3, and 6 are placed at (-3,-2), (4, 2), and
(7, -1 ), respectively, in the x, y-plane; determine the position
of the centroid of the system.
M,
x =
y =
m
2-(-3) + 3-4 + 6-7 48
11
M,
m
2 + 3 + 6
2-(-2) + 3-2 + 6-(-1)
2 + 3 + 6
j4_
11
The center of mass is at
rj& 4_
vll ' ~ 11
centroid
(48/11,^/11)
x
Analogously, for n objects with masses mj, m<i ... mn located at
n> yn> zn) *n ^ne x> y-> 2-space, we have

Section 27.1 Mass Point Systems
841
If Myz, Mzx, and Mxy denote the total moment of mass with
respect to they, z-plane, the z, x-plane, and the x, y-plane,
respectively, and m the sum total of the masses mj, m^ ... mn, the
point (jc, y, z ) of the centroid will be
(x9y9 z) =
(M,
yz
M
zx
M
XV
\ m
m
m
Three masses 2, 3, and 6 are placed at (- 3, -2, 1), (4, 2, 3 ), and
(7 , -1 , 0), respectively, in an x, y, z-space; find the position of
the centroid of the system.
M
x =
y =
yz
m
M,
xz
3-(-3) + 3-4 + 6-7 48
2 + 3 + 6 "11
2-(-2) + 3-2 + 6-(-1)
m
z =&*-
m
2 + 3 + 6
1 + 3-3 + 6-0
2 + 3 + 6
= 1
48 4
The centroid is at [ -tt- , --^-,1
J
4_
11

842
Chapter 27 CENTROIDS
27.2 Plane Figures and Laminas
A thin sheet of uniform thickness and density is known as a
lamina. If it has a geometrical center, this point will also be
its centroid (or center of mass, center of gravity, or
barycenter).
The centroid of a lamina of rectangular shape is the point of
intersection of its space diagonals.
Given a lamina of the shape shown here, find the location of its
centroid.
6
7 L2
4
12
Divide the lamina into rectangular elements as shown by the
broken lines and place it on the jc-axis of an orthogonal
coordinate system, with the y-axis at the midpoint of the bottom line.
(-4,1/2)^
V
-6
i
2
y
i
(1,4)
•
(5,2).
i
i
i
A
I (
»
X
The mass of each section is the product of the area of the
rectangle, the thickness 5 of the lamina, and the density p of
the material.
We find
x =
[4 • 1 • (- 4) + 6 • 8 • 1 + 2 • 4 • 5] dp 6
(4- 1 + 6-8 + 2-4) dp 5 '
(4 • 1 • 0.5 + 6 • 8 • 4 + 2 • 4 - 2) dp 7
y =
(4-1 + 68 + 2-4) dp
2 '

Section 27.2 Plane Figures and Laminas
843
Thus, the centroid of the lamina is at (^ 1 -, 3 ^, if placed on
the x-axis (with the midpoint of the bottom line at the origin) of
an orthogonal coordinate system. •
To obtain a more general formula for the centroid of a lamina,
consider a lamina bounded by the curves of the continuous
functions f(x) and g{x) between x = a and x=b (a<b). We
divide the interval [a, 6] into n sub-intervals with the points
of subdivision
a = xq<xi<X2 < ... <xn = b .
x
XQ=a xi X2
Let xi be the midpoint of each sub-interval, and construct on
each interval a rectangle with sides parallel to the coordinate
axes, its breadth Axt being equal to the length of the sub-
interval and its height equal to [f( x) -g(x)] :
x
XQ=a
The geometrical center of each rectangle is at ix^yi), where
_ fix) + g(x)
Ji - 2
The mass ra; of the i-th rectangle is the product of its density p,
thickness 8, and area [f(x) - g(x)] Ax;,
mi = p8 [f(x) -g(x)] Axt,
and the total mass of the lamina can be approximated to
n
m & ^p 8 [f(x) - g(x)] Axt.
i = l

844 Chapter 27 CENTROIDS
With an increasing number n of forever-diminishing rectan-
p. 1AH gles, this Riemann sum becomes a successively better
approximation of the total mass,
n
m = lim Yp 8 \f(x) - g(x)] Axi,
b
m = p8\ {f(x)-g(x)] dx,
a
which is the formula for the mass of a lamina, where f and g
are continuous functions on the interval [a, 6], p is the density,
and 8 is the thickness of the lamina.
Moment of Mass of a Lamina
The moment of mass is the product of mass and moment arm;
consequently, for the i-th rectangle the moment of mass about
the x-axis is miyi.
Since yi = and mi = p 8 \f(x) - g(x)] Ax;, the total
moment of mass ahout the x-axis may be approximated by
n
Mx * \p8f(x)+2gM lf(x)-g(x)]Axi
i = l
n
fHx)-gHx)
0 £„^_^4,..
When the sub-intervals tend to 0, this approximation becomes
an equality; that is,
n
Mx = lim > p8 5 Axt
i = l
or
Mx =^ j lf2(x)-g2(x)] dx,
a
where, as before, f and g are continuous functions in the
interval [a, b], pis the density, and 8 is the thickness of the lamina.
Analogously, the moment of mass about the y-axis of the i-th
rectangle is mi xi, and the formula for the total moment of
mass about the y-axis is
b
My = p8 J x lf(x) -g(x)] dx.
a

Section 27.2 Plane Figures and Laminas
845
For the centroid (3c, y) of the lamina, we obtain
M
PSJx[fix)-gix)] dx
X =
a
m
p8 [[fix) -g(x)] dx
a
[x [f(x) - g(x)] dx
a
[[fix) - gix)] dx
a
y
M,
m
^- j\f2(x)-gHx)] dx
a
pS [[fix) -gix)] dx
a
\ \\fHx)-gHx)} Ax
a
[[fix) -gix)] dx
a
where the denominator is equal to the area of the lamina.
Thus, the centroid of the lamina is
b
* ~ A \X ^f^ ~ Six)] dx
a
y
= YA J \f2ix) - g2ix)] dx,
a
where A is the area of a lamina bounded by x = a; x = b and the
curves of the continuous functions fix) and gix).
In the formula, the factors of density and thickness cancel out,
showing, as expected, that the location of the centroid is
independent of density and mass.
Determine the location of the centroid of the region bounded by
the parabola y = x2 and the chord y = 2x + 3 .
The points of intersection are given by
x2 = 2jc + 3
x = — 1
y = i J
x = S
y = 9J
[Not to scale]
= 2x + S
x

846
Chapter 27 CENTROIDS
The area A is
A = J (2 x + 3 - x2) dx
x2 + 3 x
X'
-.3
J-l
32
3
3
r
x =
x (2 x + 3 - x2) dx
32
J
(2 x2 + 3 x -x3) dx
3 |~2x3 3x2 x^
32 3 + 2 ~ 4
-i3
J-l
= 1
= 2T J [(2x + 3)2"(x2)2] **
-1
3
J3_
64
J (9+ 12 x + 4x2-x4)
dx
3 Tft n 0 4x3 x5
= 64 \9x + 6x 2 + —~-
-,3
_-l
17
= 3.4
The coordinates of the centroid are (1, 3.4).
[Not to scale]
= 2x + 3
x

847
27.3 Center of Mass of Solids of Revolution
While we prefer the term centroid for plane figures, the term
center of mass or center of gravity seems more appropriate
for solids.
The center of mass of a solid of revolution of uniform density
will be situated somewhere on its axis of rotation; the problem
is thus reduced to finding out just where on the axis.
We consider a solid of revolution formed when the area
bounded by the curve of fix), the x-axis, and the lines x = a and x = b
revolves about the x-axis.
y
y=fe)
>• x
A rectangular section of the area then generates a disk of
volume ny2 Ax and mass p ny2 Ax, where p is the density. The
center of mass of the disk is also its geometrical center. Since
the length of the moment arm is x, the moment of mass is
x p ny2 Ax. Letting Ax tend to 0, we find the total moment of
mass with respect to they, z-plane,
b
Myz = pn \x [fix)] dx,
a
and the total mass of the solid of rotation,
p. 839
Mx
m = pn J [fix)] dx.
a
Since x = —^—, we obtain the x-coordinate of the center of mass,
m
pn \x [fix)] dx \x [fix)] dx
x =
a
a
pn [[fix)]2 dx \[fix)]2 dx
a a
where fix) is a continuous function and a < x < b.
Revolution about the x-axis means that y = z = 0 .

848
Chapter 27 CENTROIDS
By a similar reasoning, the y-coordinate of the center of mass
formed by the area revolving about the y-axis is
y =
Jy \g(y)]2 dy
Jfe(y)]2 dy
where g(y) is a continuous function; c <y < d ; and x = z = 0 .
Determine the center of mass of a solid generated by the area
rotating about the x-axis and the y-axis, respectively, in the
first quadrant of the parabola y = 9 - x2.
4
9'
6
3
-1
-3
y
\
. y>=9
\.
I l \ l J
12 3 4
\
X
The parabola intersects the coordinate axes at
(3, 0) and (0, 9).
When the curve rotates about the jc-axis, the center of mass of the
generated solid is on the jc-axis; y = z = 0.
3
J*(9-*2)5
dx
x =
o
J(9-*2)5
dx
o
f (81 jc - 18jc3 +jc5) dx
o
[(81-18jc2 + jc4) dx
0
"81 x2
2
81 x-
9 x4 x6~
2 + 6_
JC51
6x3 + —
o
0
3
n
15
" 16*

Section 27.3 Center of Mass of Solids of Revolution
849
The center of mass is —, 0 , 0 ).
(15/16, 0, 0)
When the curve rotates about the y-axis, the center of mass of
the generated solid is on the y-axis; x = z = 0.
J y (V9 -y ) dy \y(9-y)dy
y =
o
0
J 0
0
9y2 y:
-|9
-10
-|9
= 3
9y-J-z-
jo
The center of mass is at (0, 3, 0)
(0, 3, 0)
y = 9-x2

850
Chapter 27 CENTROIDS
Find the center of mass of a right circular cone with uniform
density and of altitude h and radius r.
The cone may be generated by revolving a right triangle with
legs r and h about the leg h. We assume that h is placed along
the x-axis of an orthogonal coordinate system with the vertex of
the cone at the origin.
Similar triangles give
r
h
y t
x 9 ^ h
JX[hx
dx
x =
"tx I dx
J
0
I
h2
x3 dx
J
A2
x2 dx
, h24
h23
r2 Xs
-!■*
The center of mass of a right circular cone of uniform density
3
is on its axis, at a distance from the vertex of 7 of the height of
the cone.

851
Chapter
28
AREA
Page
28.1 Plane Surfaces 852
28.2 Surface of Revolution 860
28.3 Work 867

852
Chapter 28 AREA
28.1 Plane Surfaces
In integration of a function, areas above the x-axis count as
positive, those below the x-axis as negative; thus, integration
will give the difference between their absolute values.
To find the actual area, we must add the absolute values of the
integrals, that is, reverse the sign of the area below the #-axis;
to be on the safe side, it is good practice to plot a graph of a
function whenever in doubt.
Area of an Ellipse
The equation
— + ^1= 1
a2 b2
describes an ellipse with the axes 2a and 26 and with its center
at the origin of an orthogonal coordinate system.
x
p. 738
The area of the ellipse is evidently 4 times the area A\ of the
first quadrant. With y = —• Va2 -x2, we have
a
A L f Va2-*2
a J
o
dx
Introducing
x = a sin 0 => ya2-x2 = a cos 6; dx
we must find new limit values.
= a cos 0 dO,
x = 0: a sin 0=0
sin0 = 0
0= 0
x = a :
a sin 6
sin 6
0 = -
a
1
n
2

Section 28.1 Plane Surfaces
853
Thus,
p. 511
a
7t/2
h f I h
Ai=— \ a2 - jc2 dr = — fa cos 0 • a cos 0 d0
o o
7t/2
= ab \ cos2 0 d0
0
It/2
= — f (l + cos20)d0
0
oo T 1. ^1 nab
= — L0+2sm20Jo =-4--
Hence,
the area of an ellipse is n a 6, where a and 6 are the semi-axes
of the ellipse.
Witha = 6 = r,
the area of a circle is n r2, where r is the radius of the circle.
Area between Two Curves
fix)
g(x)
X
h(x)
b b
j f(x) dx - \ g (x) dx
a
a
jg(x) dx +
J h (x) dx
a
a
The shaded area between the curves of the functions f(x) and
g(x) and the lines x = a and x = b may be determined by
subtracting the area between g(x) and the x-axis from the area
between f(x) and the x-axis.
Similarly, the shaded area confined by curves of the functions
g (x) and h (x) and the lines x = a and x = b may be determined
by adding the area between g(x) and the x-axis to the area
between h(x) and the x-axis.
Iff and g are continuous functions on [a, 6], then the area
bounded by the two curves and the lines x = a and jc = 6 is given
by
b
\f{x) - g (x)] dx
\
a
if fix) >g (x) for all x in [a, 6] .

854
Chapter 28 AREA
Find the area between the curves
fix) = x + 1 and g ix) = x2 - 2 x + 3 .
The ^-coordinates of the points of intersection are given by
x+1 = x2-2x + 3; #1 = 2 , X2 = 1.
t
y = gix) = x2 - 2x + 3
^ = fix) = * + 1
x
fix) >gix) in the interval [1, 2] .
We have
A= f|/(*)-£(*)] dx = f[3x -x2-2] d*
~2~~ T~2*
Jl
1
6 '
Find the area between the curves
fix) = x3 - 4 x and gix) = 2x2-x3 .
The points of intersection are determined by
x ix - 2) (x + 1) = 0
x\ = 0
*2 = 2 -
*3 = -1 - '
The graph of the two functions shows that fix) > gix) in
the interval [-1, 0], while gix) > fix) in [0, 2] .
y
A
gix) -4
y = /(#) = x3 — 4x
x
y =g(x) = 2#2 — x3

Section 28.1 Plane Surfaces
855
A =
The sought area is
0 2
[[fix) - g(x)] dx + [\g(x) -f(x)] dx
0
0
= f [(x3 -4x)-(2x2- x3)] dx + f [(2 x2 - xs) - (x3 - 4 x)]
dx
o
o
= f [2 x3 - 2 x2 - 4 x] dx + f [2 x2 - 2 xs + 4 x]
dx
o
'x4 2 x3
2x2
0
J-l
2 x3 x4
TT+2^
Jo
5 16 1
7: + -r- = 6 ^ square units.
00 0
Since the beginning of 1970, the water consumption of a city
had grown as
fit) = 1.580 + 0.0020 t2,
where f(t) is measured in millions of cubic meters and t in
years.
To conserve water, better industrial use was encouraged and
certain restrictions were imposed in 1986; the next four years
(1986-89) saw a reduced growth of consumption described by the
function
git) = 1.580 + 0.0014 £2,
where g(t) is measured in millions of cubic meters.
Supposing the new trend continues, predict the total volume
saved during the ten-year period 1991 through 2000.
Since t = 0 at the beginning of 1970, we have
year 1991 => t = 21; end of year 2000 => t = 31,
and the water saved
31 31
\\f(t)-g {t)] dt = f [(1.580 + 0.0020 t2) - (1.580 + 0.0014 £2)] d*
21
31
= J 0.0006 t2dt
21
31
21
0.0002 £3 = 4.106-10 6m3.
0
(1970)
16 21 31
(1986)(1991) (2000)

Chapter 28 AREA
Area in Polar Coordinates
Just as rectangles are used to define a Riemann sum in a
rectangular coordinate system, sectors of a circle are used to
develop a formula for the area of a region whose boundary is
given by a polar equation.
Let A be an area bounded by a curve r = f(6) and the radius
vectors be at 6 = a and 6 = p. fis assumed to be a continuous,
non-negative function in the interval [a, /fl, and 0 < a, ft < 2n.
Divide the interval [a, /¾ into n sub-intervals with the points of
subdivision
a=Qo<Oi<02 <...<0n_1<Gn =p.
For each i let A0j = 6( - 6( _ i, and choose an arbitrary angle £;
within the i-th sub-interval A0j.
► 0
r=ffl
The total area is the sum of the n smaller areas, so
n
A *
i = l
n
£ |(A0;) [/(£)]2 = | £ |/(&)]2 (A<%) .
i = i i= 1
With an increasing number of points of subdivision, the
Riemann sum becomes an increasingly closer approximation
of the sought area, so
n
A= lim i£ |/(&)]2(A6|i)
n-> °° i = i
or
/J
A = | J[/(0)]2d0.
a
Find the area enclosed by the curve r = 8 cos 80, a 16-petal
rhodonea.
To determine the total area of the 16 petals, we calculate the
area of a half-petal and multiply the result by 32.

Section 28.1 Plane Surfaces
857
As 0 increases from 0, the first zero value of r occurs for 0
rad, where
r = 8 cos-77- = 8 cos— = 0 .
lb z
n
16
r = 8 cos 80
0
^W16
0=0
Let A be the area of the half-petal between the angles 0=0 and
n
0=— ; then
lb
7C/16
Tt/16
A = - I (8 cos 8 0)2 dO = 32 J cos2 8 0 dO.
0
0
Using the identity cos2 (/> = tt (1 + cos 2 0), we find
A = 32
7C/16
J*
(1 + cos 16 0) d6
0
7C/16
= 16 /(1 + cos 16 6) dO
0
,/16
= 16 0 + :^7 sin 16 0 = re .
Thus, the total area of the rhodonea is 32 n square units.
As it takes exactly 2n rad to complete the full curve of
r = 8 cos 80, the area may also be calculated by integrating
between 0 and 2rc,
2 n
J
• 82 cos2 8 0 d0 = 32 n .
0

858
Chapter 28 AREA
We have assumed until now that r = f(0), with f(O) > 0 for all 0.
If this requirement is dropped, care must be taken. In the
following problem, integration off(6) between 0 and 2n would
not give a correct answer.
Find the area enclosed by a trifolium, r = cos 3 6.
To determine the area of the three petals, we calculate the area
of a half-petal and multiply the result by 6.
We choose the shaded area in the figure below. As 6 increases
from 0, the first zero value of r occurs for 6 = n/6 rad, where
3 n n
r = cos -^- = cos 77 = 0.
6 2
r = cos 3 6
0=0
Let A be the area of the half-petal between the two angles 6
and 6 = n/6,
- n/6 1 n/6
A = - J (cos 3 6)2 d6 = - J cos2 3 6 dO .
= 0
o
0
With the identity cos2 0 = 7; (1 + cos 2 6), we obtain
- n/6
A = -j (1 +
cos 6 6) dO
o
= i [0+|sin6 0]
n/6
0
n
24*
Thus, the total area of the trifolium is
•'»-*■
As it requires only a change from 0 to n rad to complete the
trifolium, the area may also be calculated by integrating
between 0 and rc,
n
J
\ (cos 3 6)2 d6 = | .
o
Integrating between 0 and 2 n would give twice the correct
answer.

Section 28.1 Plane Surfaces
859
cardioid: heart-shaped
When, in a plane, a circle
rolls on a fixed circle of equal
size, any point of the moving
circle outlines a cardioid:
To enhance signal sensitivity in the forward direction and
limit pickup of back signals, a microphone has a
characteristic in the shape of a cardioid, r = a (1 + cos 0), where radius
vector r describes output signal level as a function of the angle
of incidence for an all-round constant sound pressure level.
Calculate the attenuation of sound emanating from behind the
K 3 K
microphone between angles 0 = 7? and 0 = -r— when placed in
an environment with a constant ambient sound level.
K
\ ► 0
The ratio of back-signal level to total pickup of the
microphone is
2a2\ \ (l + cos0)2d0
Q =
w/2
i n
2a2-r\ (l + cos0)2d0
0
J(l + cos 0)2d0= J (1 + 2 cos 0+cos2 0)d0
[l + 2cos0+2(l + cos20)J d0
■J[
3 rt „ cos 2 0"
-+2 cos 0 + —-—
d0 = ||0+2sin0+|sin2 0
The total energy emission is
Aq = a2
and the total back radiation is
Ab = a2
3k
3 1
-0+2 sin 0+ ysin2 0
2 4
-I 7t
3 na/
Jo
3 1
-0+2 sin 0+ Tsin2 0
2 4
-»n
'3k
J 7t/2
Thus,
+ C
2 a2.
q =
3k
2
2 3k
* 0.076.

860
Chapter 28 AREA
28.2 Surface of Revolution
A surface of revolution is generated when a plane curve is
revolved about an axis in the plane.
Assume that f(x) has a continuous derivative over the interval
[a, b], and divide into n intervals of equal length:
xq = a\ < xi < X2 < ... xn = b .
y=fix)
x
p. 830
p. 716
Letting A the corresponding arc length is
approximated by
As; * Vl+lT(&Xl2 Ax,,
where £; is a point in [xi _ i, xj\ by Lagrange's me an-value
theorem.
If the line segment joining (xi _ i, yi _ i) and (xi, yi) is revolved
about the x-axis, it generates a mantle surface of frustum of a
right circular cone, whose area approximates the
corresponding area, A Sj, of our surface of revolution.
radius = yi
y=fix)
x
b=Xi=n
radius/= y^i
Thus,
ASi * 2k yi~l+yi Vl+lT(&)]2 A*,

Section 28.2 Surface of Revolution
861
With Axt small, we haveyi_1 & yi & f(£i); therefore,
ASt * 2Kf(& Vl + IT(&)]2 Axi
and the total area S generated by revolution of the arc s about
the x-axis is given by
n
S= lim X2 nfiSi) Vl + 1/1½)]2 A*,
or
n —> 00 i = 1
b
S = 2n\ f(x)^l + \f'(x)]2 dx,
a
which is the area of a surface of revolution generated by
revolving the curve y = fix) about the x-axis over the interval
[a, 6], where /*' is continuous.
Find the surface area generated by revolution of the curve
y = 2 Vx about the x-axis between x = 0 and x = 2.
Iy=2/T
2
r
S = 2k
2^x
j
0
V
1 + ( j— [2 V*] ] dx
yd x
= 44^i+(*-i/2):
dx
0
= 4n J Vx v 1 + *_1 dx = 4n I Vx + 1 dx
0
0

862
Chapter 28 AREA
Let x + 1 = u; dx = du.
The new limits are:
for x = 0
for x = 2
=>
=>
u = 1
u = 3
= 471 I ^[u
du =
871
u
3/2
871
(V27-l) = 35.15 ... square units.
Find the surface area generated by the revolution about the
3/-
y-axis of the arc between x = 0 and x = 1 of the curve y = V* .
y = %Jx~
If y = V* , then x = y3.
x = 0
y = 0,
1
S = 2tc
0
X = 1
1 + |^^3]Jdy
1
= 2tc y3 V 1 + (3y2)2 dy = 271 J y3 V 1 + 9y4 dy.
0 0
Let 1 + 9y4 = u; then dw = 36y3 dy and y3 dy
The new limits are:
du
36 *
for y = 0 => w = 1
for y = 1 => u = 10

Section 28.2 Surface of Revolution
863
S = 2tc
J
1
10 10
r /— du n f i—
Vw 36 = 18 J V"
dw
K
27
10
l
^3/2 _ JL (Vl000-l) = 3.5631... square units.
27
Surface Area of a Sphere
Assume a circle of radius r with its center at the origin of an
orthogonal coordinate system, x2 + y2 = r2, and calculate the
area of the surface of revolution generated by the one-fourth of
the circumference in the first quadrant. The revolution of this
arc through a full circle generates a hemisphere:
x2 + yl - r2
(r, r, r)
x
We have v = ^1 r2 - x2 .
§=2*
r
r
J
0
Vr2 -x2 A / 1 + (-r- [Vr2 - x2\ 1 dx,
which gives
S = 4k
r
r
V^2
J
0
-*2V
1 +
-x
V
.V^2 - X2)
dx
r
r
= 4k
Vr2 - x2
j
0
Vr2 -x2
dx
= 4n J r dx = 4 7t r
0
0
x
and
S = 4KT2,
which is the well-known formula for the surface area of a
sphere with radius r.

864
Chapter 28 AREA
p. 832
Parametric Equations and Area of Surface of Revolution
Consider a plane curve given by the parametric equations
x = x(t)
y = y(t)
where a <t <b and the functions x(t) and y{t) have continuous
derivatives in [a, b].
For a curve described as a graph, we have the surface area 5,
generated by revolving the arc of f{x) between x = a and x = b
about the x-axis,
b
= 27tJ f(x)V
l + [n*)]2d*,
a
where the term yl+ \fl (x)] represents the arc length.
Having established the parametric form of the formula for arc
length,
6
r
s =
j
a
v
'dx\2
dT
;*;••
we have as a corollary
6
r
S = 2n
j
a
wf¥)
dt
as the area of a surface of rotation about the x-axis of a curve
x = x(t); y =y(t), where a<t <b, and the functions x(t) and y(t)
have continuous derivatives in [a, b].
A cycloid with the parametric equations
x = t - sin t
>
y = 1 - cos t
between £ = 0 and t = 2n rotates a full turn around the x-axis;
find the generated surface area.
x = t — sin t
y = 1 - cos t

Section 28.2 Surface of Revolution
865
2tc
r
S = 2k
(1 - cost)
J
0
V
(d ^2 fd \2
-77 [*- sin*] + -T- [1-cos*] d*
V
d*
V
dt
2tc
= 2k j (1 - cos*) Vd - cos *)2 + sin2 * d*
0
2tc
= 2 k I (1 - cost) V1 - 2 cos t + cos2 t + sin21 dt
o
2tc
= 2tc J (1 - cos*) V 2 - 2 cos * d*.
0
Writing
2tc
r
S = 2k
j
o
V1
f 1 - COS t\ n ^ II- cos * ,
. 2 A/ - d*
p. 512 and using the identity sin — =
in|=^/-
COS*
, we can write
S = 8k
2tc
r t
sin3 — d* .
o
p. 731 The reduction formula for J sin" x dx yields
sin3 — d* =
2 cos (*/ 2) sin2 (*/ 2) 2
3 3
2 cos (*/ 2) sin2 (*/ 2) 2
+ -Jsin(*/2) d* +Ci
- (2)cos(*/2) + C2.
Hence,
S = 8k
2cos(*/2) sin2(*/2) 4 , ,rtX
- -cos(*/2)
-,2tc
0
64k
* 67.02.

866 Chapter 28 AREA
Guldin's First Rule
Using arc length and center of mass, the formula for the
surface area of a surface of revolution,
b
S = 2n \ f(x) ^1+ [fix)]2 dx,
a
may be expressed:
The surface area of a surface of revolution is equal to the
product of the arc length of the generating curve and the
length of the path described by its centroid.
In 1615 Johannes Kepler - the eminent German astronomer
and mathematician - described and proved this theorem, later
known as Guldin's first rule after Swiss mathematician Paul
Guldin, who published it in his four-volume Centrobaryca in
1635 - 41.
The theorem was already known by Pappus of Alexandria,
whose Synagoge ("Collection") was published around A.D. 340
in twelve books, of which the first, part of the second, and the
last are lost. The Synagoge is considered one of the best
sources of information on ancient Greek mathematics.
Surface Area of a Torus
A circle of radius r rotates around an axis in its plane at a
distance R from the center of the circle, generating the torus
shown here.
We have:
1. The centroid of the circular area is at its center, at the
distance R from the axis of the generated torus; the
distance traveled by the centroid is 2 n R.
2. The circumference of the circle is 2 n • r.
According to Guldin's first rule, the surface area of the torus is
S = 2nr- 2nR = 4n2rR.
Guldin's First Rule
KEPLER Johannes
(1571-1630)
GULDIN Paul (Originally,
(1577-1643) Habakkuk)
PAPPOS
(c. 300)

867
28.3 Work
JOULE James Prescott
(1835 -1889)
In natural sciences, statistics, and economics, applications of
the determination of area under a curve are legion. As an
example we choose to determine work done by a variable force.
If an object moves along a straight line, then the force F acting
on an object of mass m — in the direction of the movement -
d2s
will give it an acceleration a = —— (Newton's second law of
dt2,
motion); thus, if distance is s and time is t,
d2s
F = m
d*2 '
In the SI (Systeme international d'unites), mass is measured
in kilograms (kg), distance in meters (m), time in seconds
(s), and the force in newtons (N), and 1 N = 1 kg ■ 1 m/s2.
Thus, acting on a mass of 1 kg, the force of 1 N produces an
acceleration of 1 m/s2.
The work (or energy) W is the product of the force F and the
distance s that the object moves; thus, W = F • s.
In SI, the unit of work is the joule (J), named after the English
physicist James Prescott Joule. 1 J = 1 N • m (newton • meter).
Lifting a mass of 3.20 kg straight up 0.50 m against the
acceleration of gravity of 9.81 m • s-2 requires a force
F = 3.20 [kg] • 9.81 [m • s"2l = 31.39 N,
and the work produced is
W = 3.20 [kg] • 9.81 [m . s"2] • 0.50 [m] = 15.70 J.
Consider a variable force fix), where f is a continuous
function, acting on an object, moving it along the x-axis in the
positive direction:
Force
F
A
a= xq x\ X2
F = fix)
x Distance
*3 b = *4
Riemann sum: p. 747
As before, we subdivide an interval [a, b] into n sub-intervals
of equal length with the points of subdivision
a = Xq<X\<X2 < ... <xn =b
and find

868
Chapter 28 AREA
Xq - a, Xi = a +
a
X2 = a +
2 (6-a)
x
n
a +
n{b -a)
n n n
Within the i-th sub-interval, choose an arbitrary point (¾
^! in fro, *i], £2 in [*i, x2], ..., ^ in bn_i,acj
= b
Force
F
A
a= Xq
X\
F = fix)
Si
*2
<?2 £3
*3
x Distance
6=^4
$4
A rectangle whose base is the sub-interval [*£ _ 1, #;] and whose
height (representing force) is f (^) corresponds to an
increment of work Wi; the total work done represented by the
area A under the curve is the sum of all rectangles Wi,
n
W= A * I /(£)(*i-*i_i),
i = l
or
n
W * I f(b)AXi .
i = l
With an increasing number of points of subdivision, this sum
becomes a forever-closer approximation of the total work
produced,
W = lim £ f(ti)*Xi
n—> 00 i = 1
or
b
W = \f(x)dx .
a
A force F (x)
16
x + 3
N, where # denotes the distance in meters
from the starting point, moves an object along the x-axis.
What is the total work, in joules, expended in moving the
object from x=ltox = 4?
W =
4
r 16
x + 3
dx = 16
In (a + 3) * 8.95 J

Section 28.3 Work
869
A spring requires a force of 42 N to stretch it from its natural
length of 0.12 m to 0.16 m.
By Hooke's law - named for the English mathematician and
scientist Robert Hooke - the force required to stretch (or
compress) a spring is proportional to the change in length.
Calculate the work required to stretch the spring from 0.16 m to
0.20 m.
///////////////////////////////////
T
0.12 m
0.16 m
0.20 m
Let f (x) express the force as a function of the distance the
spring is stretched from its equilibrium position,
f(x) = k • x, where k is the proportionality constant.
f (0.16 -0.12) = k- 0.04 = 42; k = 1050.
The work required to stretch the spring from 0.16 m to 0.20 m is
0.08
W
■J
0.04
1050 x dx = 525
0.08
0.04
x2 = 2.52 J.
Totentia Reflitutiva,
O R
SPRING.
He Theory or Springs, though
attempted by divers eminent
Mathematicians of this Age
has hitherto not been Publifhed
by any. It it now about eighteen
years fince I firft found it our,
but designing to apply it to Tome
..... , particular ufe, I omitted the
publilhing thereof.
About three years fince His Majefty was pleafed to
fee the Experiment that made out this Theory tried
zt\\htfe-H*U,M alio my Spring Watch.
About two years fince I printed this Theory in an
Anagram at the end of my Book of rheDefcriptions of
Hchofcopes,i//fc.r ciiino s ssttu ujd eftjjt te»(to fc
iW}Thatis, The Power of any Spring is in the fame
proportion with the Tenfion thereof: That is, if one
power ftretch or bend it one fpace, two will bend it
two, and three will bend it three, and fo forward.
Now as the Theory is very fhort, fo the way of
trying it is very eafie.
In his introduction to
Potentia Restitutiva
(1678), Robert Hooke
points out that about
two years before, at
the end of another
publication, he had
printed, as an
anagram, ut tensio sic vis
to ensure the priority
right of his discovery
of the proportionality
between force applied
and the length of a
spring.

870 Chapter 28 AREA
From Georg Reisch, Margarita phylosophica (first edition, 1496; here
reproduced from a 1583 reprint).
The seven categories of liberal arts were, in the Middle Ages,
personified by women. Dame Geometry represented "the
science of unchangeable shape and form". The illustration
points out the practical use of geometry in masonry, carpentry,
surveying, and astronomy.
Among the seven liberal arts, the quadrivium [geometry,
astronomy, arithmetic, and music] constituted a division to
which the trivium [grammar, dialectics (logic), and rhetoric]
was subordinate.

871
Chapter
29
VOLUME
Page
29.0 Introduction 873
29.1 Disk Method 873
29.2 Shell Method 877
29.3 Solids Generated by Area Bounded by Two Curves 880
29.4 Translation of Axes 882
29.5 Guldin's Second Rule 884
29.6 Solids with Known Cross-Section Areas 885
29.7 Transforming Double Integrals from Orthogonal to
Polar Coordinates 887

872
NO VA
STEREOMETRIA
DOLIORVM VINARIORVMj INPRI-
mis Auftriaci, figurx omnium
aptifsims;
z t
USUS IN EO VIRGiE CUBI-
cx compendiofiffimus &
plane fingularis.
Acceflic
STEREOMETRIiE A R CHIME-
dex Supplementum.
Authorc
Ioanne Kepplero, Imp. Cxf. Matthise I.
cju% fidd. Ordd. Auftnz fupra Arufum
Mathcmatico.
A a it * % wEKdB I WJiJ M.dc.xk
LlNC I I
Zxosdtixu JOANNES PLANCVS, fumotibus Author*.
Johannes Kepler, "New Solid Geometry of Wine Barrels, Especially
Austrian Barrels ..." (1615).
Stocking his wine cellar in 1613, Kepler distrusted the wine
merchants' crude methods of volume estimation; he
challenged them with computations, which he published in Nova
stereometria.... He computed the volume of solids obtained by
the rotation of figures about a line in their plane. Kepler's
mathematical trials were of decisive importance for the
development of integration methods of calculus.
Kepler's text is also remarkable as one of the first to use
methods of differential calculus in the study of maxima and
minima. Kepler states: decrementa habet insitio insensibilia
("near a maximum the decrements on both sides are in the
beginning only imperceptible"). [D.J. Struik, A Source Book
in Mathematics 1200 - 1800; Princeton University Press, 1986.]

873
29.0 Introduction
To determine the volume of a solid of revolution by
integration, one may think of the solid either as consisting of an
infinite number of disks with their central axis coinciding
with the axis of revolution or as formed by an infinite number
of concentric layers; the methods used are called the disk
method and the shell method, respectively. A third method is
based on the area of the generating figure and the path of the
centroid.
With cross-section areas known, the volume of a solid of any
form, whether or not a solid of revolution, can be determined.
29.1 Disk Method
The area under a curve can be determined by dividing it into
rectangular strips and summing the areas of all the strips as
the number of partitions tends to infinity.
If we want to determine the volume of a solid of revolution
formed by revolving the curve of a continuous f(x) about the x-
axis, we may divide the solid into a series of disks with
thickness Ax, radius /"(*;), and volume n\f(xi)] Ax, where xi
is the endpoint of a sub-interval on the x-axis and the disks are
placed like pineapple slices on the axis of revolution.
If n is the number of disks on the x-axis and we let n tend to
infinity, then the volume of revolution about the x-axis is
given by n
V= lim £ti \f(xt)]2Ax
or, in integral form,

874
Chapter 2 9 VOLUME
V = 71 f \f(x)]2 dx,
a
which is the volume of a solid of revolution generated by
rotating the region bounded by the graphs of y = f(x), y = 0, x =
a, and x-b about the a>axis.
Analogously,
V = n J lg(y)]2 dy
is the volume of a solid of revolution generated by rotating the
region bounded by the graphs of x = g(y), x = 0, y = c, and y = d
about the y-axis.
Find the volume generated by the part of the graph y
between x = 1 and x = 2 when rotating about the x-axis.
= x2
x
-3 -2 -1
12 3
The sought volume is
V
= 7i J (x2y
dx = •=•
K
5
jc5 =
31 K
5 '
= v2 T.Q
y =x
X

Section 29.1 Disk Method
875
Find the volume generated by the part of the graph y = x2
between y = 1 and y = 4 when rotating about the y-axis.
x
Since the curve revolves about the y-axis, we write the equation
y = x2 in the form x = Vy • The volume is
/* o 4
V = 71 I (Vy) dy = 71 J y dy = ~
4
1
y2 =
15 n
y
t

876
Chapter 2 9 VOLUME
Ellipsoid of Revolution
An ellipsoid of revolution is generated by rotating the ellipse
around the x-axis.
x2 y2
a2 b2
To obtain the volume of the ellipsoid of revolution, we
determine the volume generated by the arc; the volume is obtained
by integrating the area of the disks from x = - a to x = a. With
y = — ya2 - X'
J a
we obtain
a
r
V = 71
J
-a
-i2
71 b2
a
a
ya2 - x2
dx
a'
J (a2 - x2) dx
-a
Kb2
a'
Q-lO
y O
az x - —
4 nab2
-a
which is the volume of an ellipsoid of revolution generated by
an ellipse of axes 2a and 26 rotating about its major axis 2a.
If we look upon a sphere of radius r as an ellipsoid of
revolution with all axes equal,
a = b = r,
we obtain the formula for the volume of a sphere:
4 7tr3

877
29.2 Shell Method
In the shell method, the volume of a solid of revolution is
calculated as the sum of the volumes of an infinite number of
concentric cylindrical shells placed concentric with the axis
of revolution:
If we let xi be the endpoint of a subinterval on the x-axis, then
the approximate volume of one cylindrical shell is the product
of the circumference of the outer edge of the shell, its height
f(x{), and its thickness Ax.
Working on the positive side of the x-axis, divide the solid into
a series of cylindrical shells with circumference 2nxi > height
f{xi), thickness Ax, and volume 2nxi \f(x0] &x > where xi is the
endpoint of a sub-interval on the x-axis. If n is the number of
partitions of the x-axis of the solid of revolution, and we let n
tend to infinity, the volume of revolution about the y-axis is
n
V = lim V27iXj \f(X()] Ax
n ->°° i = 1
or, in integral form,
6
V = 2ti J x fix) dx,
a
which is the volume of a solid of revolution generated by
rotating the area between the graphy = fix) and the x-axis and
between x = a and x = b ahout the y-axis.
Analogously, working on the positive side of the y-axis, we
have
d
V = 27i J y giy) dy,
c
which is the volume of a solid of revolution generated by
rotating the area between the graph x = giy) and the y-axis and
between y = c andy = d ahout the a>axis.

878
Chapter 29 VOLUME
A region bounded by the graphs of y = x2, y = 0, x = 1, x = 2
revolves about they-axis; find the volume by the shell method.
x
V = 27i J x (x2) dx = 2n \ x3 dx =
71
X
4 _
15 K
2 '
y - 3 x3 - 4 x2 - x + 5/2
^1
The region bounded by the graphs ofy = 0,x = 0,x = q*, and
5
y = Sx3 - 4x2 - x + o" revolves about the y-axis; find the volume
generated.
/
1/3 2/3 1 4/3
x
y =3x3-4x2-x + 5/2
^^&
The equationy = 3x3-4x2-x + 5/2 cannot be rewritten to give
x as a function of y and, consequently, is not amenable to the
disk method. On the other hand, the equation is suitable for the
shell method.

Section 29.2 Shell Method
879
4/3
r
V= 2ti
j
0
x I 3jc3 - 4x2 - x + — I dx
4/3
= 271
3x4 - 4x3 - x2 +
5 jc
cbc
o
= 27C
'3x5
X'
T+~
214/3
0
87C
5
The decision as to which method to use for finding the volume
of a solid of revolution depends on the shape and position of the
bounded region relative to the axis about which it is to be
revolved.
A region bounded by the graphs of y = x2,y = 0, and x = 2
revolves about the y-axis; find the volume generated.
With the shell method,
V= 2n \ x(x2) dx = £
0 Z
x4 = 8tc
0
x
-^

880
Chapter 2 9 VOLUME
29.3 Solids Generated by Area Bounded by
Two Curves
The volume of a solid generated by the revolution of an area
bounded by two curves is determined as the difference of the
volumes generated by the areas bounded by each of the curves
and the axis of revolution.
3/~ 9
A region bounded by the graphs of y = yx and y = xz revolves
about the x-axis; find the volume generated.
y
y = x2
y=^x
x
The curves intersect at x = 0 and x = 1.
x
y
y =2/F
X
X
To find the volume of the solid, subtract the volume generated
by revolving the area B between the curve y = x2 and the x-axis
from the volume generated by revolving the area C between
3/~
the curve y = ^ x and the x-axis.
We have, by the disk method,
V
= 71 I \V* J dx-K J (x2)'
dx
0
0
= 71 f (x2/3-x4) dx = | [3*
5/3
X5]
0
2n
5 '
o

Section 29.3 Solids Generated by Area Bounded by Two Curves
881
The area between the graphs of
y = x3 + 2 ; y = 3x + 4
and the y-axis revolves about the y-axis; find the volume
generated.
x3 + 2 = 3x + 4 => x3 - 3 x - 2 = 0 => (x-2)(x + l)2 = 0,
and thus x = 2 is one solution to the equation.
One point of intersection between the graphs is at x = 2.
y = x3 + 2 intersects the y-axis at y = 2, and y = 3x + 4aty = 4.
x 0
► x 0
0
y = xs + 2
y = 3 x + 4
■vP
x
To find the volume of the solid, subtract the volume generated
by revolving the area C between the curve y = x3 + 2 and the
y-axis from the volume generated by revolving the area B
between the curve y = 3x + 4 and the y-axis.
Applying the shell method, we have
2 2
V = 2 7i J x (3 x + 4) dx - 2 n \ x (x3 + 2) dx
0 0
2ti J (3x2 + 2x-x4) dx = 2n
0
3/y*0
O
2 _ 56 K
o" 5

882
Chapter 2 9 VOLUME
29.4 Translation of Axes
If a region in the x, y-plane revolves about a line parallel to the
x-axis, the volume of the resulting solid of revolution may be
determined after translating the region and the line to make
the axis of revolution coincide with the x-axis; similarly for
an axis of revolution parallel to the y-axis.
• A region bounded by the graphs of y = -x2 and y = - x revolves
about
a:theliney= -2; b: the line x = - 2.
Find the volumes generated.
The points of intersection are given by
-x2 = -x ; x = 0 and x = 1 .
a:
To make the axis of rotation coincide with the x-axis, translate
the graphs:
y = -x2 becomes y = 2-x2
y - -x becomes y = 2-x
y
A
axis of_g
x
revolution
y -2-x
y
A
y = 2-x
2
axis of
-1
revolution
x

Section 29.4 Translation of Axes
883
Applying the disk method, we obtain
= 7C J (2-
0
x2)2 dx - 7i J (2 - x)2 dx
0
7i J (x4 - 5 jc2 + 4 jc) cbc = 7i
0
15 *
2x2
J 0
b:
x
= -X
X
y = -(*-2)
y = -(jc-2)2
V =
Translate the curves of the graph to the right by two units to
change the domain of interest from [0, 1] to [2, 3]:
y = —x2 becomes y = -(x-2)2
y = -x becomes y = - (x - 2)
Applying the shell method, we obtain
3 3
2 7C [ x [- (x - 2)] dx - 2 n [*[-(*- 2)2]
dx
2 k J (x3 - 5 x2 + 6 x) dx
2
2%
-+—+3x2
13
5n
6

884
Chapter 2 9 VOLUME
29.5 Guldin's Second Rule
Guldin's Second Rule
GULDIN Paul (Habakkuk)
(1577-1643) p. 866
Pappus's Theorem
PAPPOS
(c. 300)
p. 845
The volume of a solid of revolution is equal to the product
of the generating area and the length of the path
described by its centroid.
This rule is commonly referred to as Guldin's second rule,
named for Paul Guldin and, like the first rule, published
in his Centrobaryca (1635 - 41). The rule is equally well
known as Pappus's theorem after Pappus of Alexandria. As
with Guldin's first rule, Kepler proved Guldin's second rule
about 20 years before it was published by Guldin.
We can verify the rule in the case of a solid generated by the
rotation around the y-axis of an area contained between the
non-intersecting curves f(x) and g{x) and the lines x = a and x
= 6, both on the same side of they-axis. Using the shell method,
we have
6
V = 2 71 \x [f(x) - g(x)] dx ; 0<a <b and f(x) > g(x).
a
If A is the area and x the x-coordinate of the centroid, then
b b
5c = -j- x [/*( *) - g (*)] dx ; hc[/*(x)-^(x)] dx = ~x A;
a
a
that is,
V=2nxA,
where 2 n x is the length of the path described by the centroid of
the area A generating the solid.
Volume of a Torus
A circle of radius r generates a torus by completing one
revolution about an axis, located in the plane of the revolving
circular area, at a distance R from the center of the circle.
We have:
1. The centroid of the circular area is at its center, at the
distance R from the axis of the generated torus; the
distance traveled by the centroid is 2 n R.
2. The area of the circle is 71 • r2.
According to Guldin's second rule, the volume of the torus is
V = 2nR(nr2) = 2n2r2R.

885
29.6 Solids with Known Cross-Section Areas
The volumes of solids of any shape whose cross sections are
known may be determined by integration,
V = J Ai(x) dx,
a
where V is the volume andAi(x) is the area of a cross section
at x, and
a
V= \A2(y)dy,
where V is the volume and A2iy) is the area of a cross section
aty.
Find the volume of the solid whose base is the ellipse
x2 y^_ =
16 + 9
and whose cross sections - perpendicular to the x-axis
squares.
The graph of the base of the solid is
- are
*i+^ = l
16 9
x
Solving for y,
y = 3
V 16
Let the height of the solid at x be h; since the cross section is a
square,
h = 2
-3^^
and the area of the cross section is h2; that is,
1 2
A(x) =
6
a/ 1-*
2
16J
4
r
V =
6
2
16J
dx, or, because of symmetry:
-4

886
Chapter 2 9 VOLUME
4
V =12
1-
j
-4
V
16 )
dx = 72
x -
48
4
0
= 192
6
The horizontal cross section of a solid is an equilateral
triangle whose side length is the square of its distance from the
topmost vertex of the solid; the height of the solid is 2 length
units. Find the volume of the solid.
The area A of the equilateral triangle is
[(2-;y)2]2V3
A =
4
and the volume V of the solid is
2
square units,
V = J dy = — \3 cubic units.
0

887
29.7 Transforming Double Integrals
from Orthogonal to Polar Coordinates
p. 753
p. 583
x
Double integrals over regions such as circles, ellipses,
cardioids, and petal curves are generally more conveniently
evaluated in polar form than in orthogonal form.
We are familiar with the conversions
x = r cos 0; y = r sin 0; r = V*2 + y2 •
The formula for transforming double integrals from
orthogonal to polar coordinates is
J J f(x,y) dx dy = J J fir cos 0, r sin 0)r dr dO,
R
xy
R
re
where f(x, y) is a function over the region Rxy of the x, y-plane
in an orthogonal coordinate system, corresponding to the
region R rQ of the r, 0-plane in a polar coordinate system.
By studying the figure, we can accept the presence of the factor
r in the polar form of the integrand. If r is the radius vector,
and 0 is the central angle in radians, then the length of the
larger arc of the sector is r A 0, and the shaded area A R is
approximated by
AR * rAOAr .
We shall solve the following problem with and without
transformation to polar coordinates. The former method
demonstrates a greater simplicity; the latter affords an
opportunity of using several interesting integration techniques.
Find the volume V bounded by the cylinder x2 + y2 = 1, the
x
x, y-plane, and the graph of the function fix, y) = — + y3 + 2.
A z = f(x, y) = x/2 + y3 + 2
x2 + y2 = 1

888
Chapter 29 VOLUME
X
1. Solving in Polar Coordinates
The radius r of the given cylinder is 1; thus, in the polar
coordinate system the lower limit of r is 0, the upper 1. A
complete revolution of the radius vector corresponds to an
angle of 2n rad; thus, we can take the lower limit of 0 to be 0,
the upper to be 2n.
p. 583 Since x = r cos 0 and y = r sin 0, we have
e =27t
r=l
V= J
e =o
j
r = 0
r cos 0
+ (r sin 0)3 + 2
r drdO
6 =2n
r=l
I
e =o
j
r = 0
(r2 cos 0
V
+ r4 sin3 0 + 2 r dr d0
sin2 0 + cos2 0=1
sin2 = 1 - cos2 0
Thus, sin3 = (1- cos2) sin 0
2k
= J
0
r3 cos 0 r5 sin3 0
6
+ r
r = l
r = 0
d0
2*f cos 0 sin3 0
■ (■
0 V
+ 1 d0
2/ f cos 0 (1- cos2 0)sin 0
27C x
+ i Ua
cos 0 sin 0 cos2 0 sin 0 . ,^
— +— ~ 6 +Ud°
— • sin 0--- cos 0 + — • cos05 0+0
o 5 15
27C
0
= 271
x=Jl^y2
X
"11
'""<x = -fT^
2. Solving in Orthogonal Coordinates
y has the lower limit -1 and the upper limit 1,
Solving x2 + y2 = 1 for x yields
x = ±Vi -y2 •
= 1 JC=Vl-V2
V =
x \
V
y = -i * = -VTTy2
^ r 2
J
-1
__l .,3
— + ydx + 2jc
- JC= -Vl-V2
dy

Section 29.7 Transforming Double Integrals to Polar Coordinates
889
-i
l-Jil + yz vTTyr + 2 vTT^
l-v2
4
- y3Vl-y2 - 2Vl-v2
dy
-1
1 1
= 2 J y3 Vl-y2dy + 4 j Vl-y2 dy ,
-1 -1
which are solved separately.
Substitute v = sin 9; Vi - y2
= cos 0, dy = cos 0 d0, and the
new limit values are
for y = - 1
fory = 1
1 = sin0; 0 = -|
1 = sin 0; 0 = —.
-1
7C/2
J«
-7C/2
2 J y3 Vl-y2dy = 2 J sin3 0 cos 0 cos 0 d0
sin2 0 + cos2 0=1
sin2 = 1 - cos2 9
Thus, sin3 = sin 0(1 - cos2)
7C/2
= 2 J sin3 0 cos2 0 d0
-7C/2
7C/2
= 2 J sin 0 (1 - cos2 0) cos2 0 d0
-7C/2
7C/2
= 2 J sin 0 (cos2 0 - cos4 0) d0
-7C/2
= 2
cos
3 0 cos5 0'
7C/2
-7C/2
= 0.
7C/2
4 Vl -y2 dy = 4 J cos 0 cos 0 d0
_1 -tc/2
7C/2 7C/2
= 4 J cos2 0 d0 = 2 J (1 + cos 2 0) d0
-7C/2 -7C/2
Thus,
7C/2
[2 0 + sin 2 0] = 2 71
-7C/2
y = 0 + 271.

890 Chapter 29 VOLUME
From Georg Reisch, Margarita phylosophica (first edition, 1496; here
reproduced from a 1583 reprint).
Authors and illustrators of the Middle Ages used poignant
scholastic methods. The above illustration is not from a book
on warfare or trauma care, but deals with geometry and is
accompanied by the following text:
- "What is a solid?
- It has length, depth, and breadth. ... Imagine a spear
that penetrates the head and reappears at the posterior; it
measures the length. Another spear pierces the chest and
emerges from the baefc it measures the depth. A third one
passes through the body from one side to the other; it
measures the breadth.
[Anders Piltz, Die gelehrte Welt des Mittelalters, 1982 J

891
Chapter
30
MOTION
Page
30.1 Laws of Kepler and Newton 893
30.2 Differentiating Distance and Velocity 896
30.3 Integrating Acceleration and Velocity 900
30.4 Velocity Vectors and Acceleration Vectors 901
30.5 Space-Time; Mass and Energy 904
*** 34.3 Fall through a Resisting Medium 1007
Oscillating Motion 1008
*** Indicates cross-reference

892
\*
» •*.
V
Carl Milles (1875 - 1955), Pegasus.
© Photo: Sydsvenska Dagbladet

893
30.1 Laws of Kepler and Newton
ZENON
(c. 490 - c. 435 B.C.) cf. p. 275
GALILEI Galileo
(1564-1642)
To make us ponder the relation between the finite and the
infinite, Zeno of Elea proposed that a soaring arrow in actual
fact stands still. At any specific instant, Zenon argued, the
arrow occupies a volume of space precisely equal to that of
the arrow; if the arrow indeed moved, it would - in every
instant - occupy a volume larger than its own and, equally
absurd, every instant would have two portions - a foregoing
and a succeeding one.
This paradox and modern physics concur that the arrow is at a
specific position at any specific instant, and that there is no
inherent difference between an arrow at rest and in motion.
So, where is the fallacy?
The difference between rest and motion manifests itself
only when we look at position at different points in
time.
If - at different points in time - the arrow is at the same
position, it is at rest; if at different positions, in motion.
The old belief that the acceleration of a freely falling body is
proportional to its mass was refuted by Galileo Galilei. In his
Discorsi e Dimostrazioni Matematiche (1638), Galileo lets
"Simplicio" defend the old idea that a body of greater mass
falls faster than a body of lesser mass, while "Salvati"
develops the doctrine that they have equal velocities.
DISCORSI
DIMOSTRAZIONI
M A T E M A T I CHE,
inform a due nuoucfiien^e
Attcncmi alii
Mecanica £ci Mo vim en ti Locals
Hd Sign&r
GALILEO GALILEE LINCEO,
Fihiibtbc MaCfimaticopriiiuriotJdSercniilimii
Grand Duca Hi Tofcjju.
Can vnn AfttndittddttntTtitgrmitAttdMui Selidt
IN I. E I D A.
AjiprcnbgliEKivira «. * c. xxxviu.
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di ciafiheduno corpo graut cadtnttJU vna da natvra determinate
velodta \fiche iaccrcjctrglicla, $ dtmtnuirglida nmfipoffafi non
ton vfirgli vioienz>atb fipporg/i qudcht impediments
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ft rnnouaper tfimpio con ottogradsdt vchtita, & vnx minore con
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fimuoua piit vefetemente dtt mtngrauejo vi conclude ilpik #rauc
muottcrji men fvetocemente.

Chapter 30 MOTION
PTOLEMAEUS Claudius
(c. 100-c. 168)
COPERNICUS Nicolaus
KOPERNIK Mikolaj
(1473 -1543)
KEPLER Johannes
(1571 -1630)
BRAHE Tycho
(1546 -1601)
NEWTON Isaac
(1643 - 1727)
In Discorsi ... (previous page), Simplicio accepts that if two
bodies of different mass have varying velocities at free fall
then - tied together - the body with the greater mass should be
delayed, while the one with the lesser mass should be hastened.
Salvati declares that he has thus made Simplicio conclude that
a body with a greater mass sometimes can fall more slowly
than a body with a lesser mass.
In ancient Greek astronomy and geometry, the circle
represented periodic or repetitive motion; planets were believed to
move in circular orbits with the Earth as center - the
geocentric system. Such orbits could, however, not explain
why, at times, planets seemed to move backward. Ptolemy
explained such retrograde motion by superimposing small
circles - epicycles - on the original circle.
The idea of circular orbits and epicycles was retained by
Copernicus, who introduced the heliocentric system, that is, the
model of the solar system centered on the Sun, with the Earth
and the other planets moving around it.
After Johannes Kepler's description of planetary motion,
founded primarily on Tycho Brahe's observations, there was
no further need to use the concept of epicycles. Kepler's three
laws of planetary motion read:
1. All the planets of the solar system describe elliptical orbits,
having the Sun as one of the foci.
2. A radius vector joining any planet to the Sun sweeps out
equal areas in equal periods of time.
3. The squares of the periods of revolution of the planets about
the Sun are directly proportional to the cubes of their mean
distances from the Sun (the major semi-axes of the
elliptical orbits).
Newton developed calculus to account for Kepler's laws.
Newton's laws of motion describe the effect of external forces
on the motion of a body; the same laws of motion apply both to
the planets and to motion here on Earth.
Newton's three laws of motion read:
1. A body is at rest or moving at a constant speed in a straight
line unless a force compels it to change that state.
2. The force on a body is equal to the product of its mass and
acceleration.
3. The actions of two bodies upon each other are always equal
in magnitude and opposite in direction.
Four years before Newton was born, the first law had been
demonstrated experimentally by Galileo. Thus, Galileo
understood that force was needed to change motion, not to sustain
constant motion. He demonstrated that the Earth's gravity
exerted constant downward acceleration, which he measured
to be - here in modern units - 9.8 meters per second per
second.

Section 30.1 Laws of Kepler and Newton
895
The planets are held in their orbital course by the gravitational
force produced by the Sun. Newton showed that celestial bodies
need not always follow the elliptical courses specified by
Kepler's first law but can take paths defined by parabolas or
hyperbolas; thus, an object with sufficient energy to surpass
the gravitational forces - e.g., a comet - can enter the solar
system and depart again.
From Newton's second law all of the fundamental equations of
dynamics can be derived by methods of calculus.
Kepler never numbered the laws that now bear his name
or specially distinguished them from his other discoveries
on planetary motion. Neither did Newton number the laws
of motion in Philosophiae naturalis principia mathematica
(1687) in the way we know them. In the facsimile below,
"Def. 111" is what we now call Newton's first law, or the law of
inertia.
^■■'ih'lhlM I*!* ■■!■!■■■
' '
PHILOSOPHISE
NATURALIS
PRINCIPIA
MATHEMATICA
Prole. (Tort Lmv/tw^ & Sodrtshs IVijjfa SoJiliu
IMPRIMATUR-
L 0 K D 1 K t,
Jiifli* Skkmv Rr*tf sc Tyjw l^f*" Stnsttr VnAu tyuA
Dcf. in.
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ttttttittfc "ft) perfcverjt itt Jlattiftto wl tjuiefccttdi <vd ttxwctidl
iwiforttttter itt dtre&Httt.
Kacc fcmpcr proporrionalis eft fuo ccrpori, ncq-5 cliflcrt quic-
quamab incrrh Mafia:, nil! hi modo enncipiendi. Per iiicrtiam
marerix fir uc corpus omnc dc ftatu fuo vcl quiefccr.di vcl moven-
didiuicffltcr dcturbctur. Unde ctiain visinfita nomine figiiifican*
tiflimo vis iucrtis die! poflfit. Excrcet vcro corpus hanc viin (ohmi-
modo in mutaiione fcatusfui per vim aliam in fc iiriprcilhm fafbi,
cftq', excrcirium ejus fubdivcrfo rcTpccliicc Rcfiftciitia ct Impetus :
Rcfiftcfftia quatcnus corpus ad confcrvandiim (latum fumn rchicl-
atur vi imprcflx; Impetus quatcnus corpus idem, vi rcfiftcntis ob-
(laculi difTicuItcrccdcndo.cohaturfiatuni ejus imirarc. Vulgus Rc-
(xftcntiam quiefcenribus ct Impetum moventibus tribiut; fc-d
moras et qiiics, utivulgo concipuinrur, refpcclufbio diftinguuntur ab
inviceni, neq^ fcmpcr verc quiefcunr qux vulgo tanqxiam quiefcen-
tia fpeftantur.
Newton developed and used calculus in his preparation of the
Principia - the fundamental work for the whole of modern
science - but he did not use it in the publication; if the book had
been presented with calculus, Newton's contemporaries would
have understood the text poorly.

896 Chapter 30 MOTION
30.2 Differentiating Distance and Velocity
The derivative of a function with respect to an independent
variable describes the rate of change of the function when the
independent variable changes. An illustration from physics
is provided by the motion of an object - which may be
rectilinear, if the object moves along a straight line, or
curvilinear, if the path of movement is curved.
The simplest form of rectilinear motion is the free fall of a
body under the influence of its own weight, which is the product
of its mass m and the acceleration of gravity, g. The velocity
of the fall increases continually - that is, the movement
accelerates - until the body reaches the end of the drop where
its kinetic energy is converted into other forms of energy,
predominantly heat.
Thus, three fundamental aspects of motion can be
distinguished: acceleration, velocity, and distance.
Acceleration is proportional to the force acting on the body;
mass times acceleration is equal to force.
When there are no external forces, a body remains at rest or
continues to move along a straight line at a constant velocity;
thus, its acceleration is zero.
A body's weight is equal to its mass times the value of the
gravitational acceleration. While the mass of a body is
constant, its weight varies with the local value of the gravitational
acceleration.
Units of Measurement
The International System of Units (SI) expresses distance in
meters (m), velocity in meters per second (m/s or m • s_1), and
acceleration in meters per second per second or meters per
second squared (m/s2 or m • s~2).
Distance
Position Distance, or position, s, is the sum of a body's consecutive
displacements along the path of rectilinear motion, that is, the
total length of the path traveled during the period of time
studied.
Velocity and Speed
Velocity, v, is the rate of change of the position of a body in the
direction of movement; that is,
velocity is the first derivative of distance with respect to
time, <js
v = -jt = s ' (t) = s ,
Rectilinear Motion
Curvilinear Motion
Force
F = ma
Weight

Section 30.2 Differentiating Distance and Velocity
897
where the dot is used in place of the conventional prime
to distinguish a derivative with respect to time; Newton
used the notation s and called it the fluxion of the fluent s.
Velocity has direction; if the velocity is considered positive for
an object moving in one direction, it is zero when the object
reverses its direction and negative when the object moves in
the reverse direction.
For v = s to be true, s must be the directed distance from any
fixed point (positive in positive direction, negative in negative
direction).
Speed is distance passed per unit of time; it is the absolute
value of velocity.
instantaneous
velocity V{
sn —
Q
Sp
average
velocity ua
Average Velocity
Instantaneous Velocity
In a distance-time graph - above - the slope of the chord PQ
represents the average velocity va of the moving body over the
range from P to Q; the instantaneous velocity V{ at the point P,
generally called only velocity, is the slope of the tangent at P.
^a =
change in position sq - sp
time ~ tq-tp
Acceleration
Instantaneous Acceleration
Average Acceleration
Acceleration, a, in its turn, is the rate of increase of velocity,
that is,
acceleration is the first derivative of velocity and
the second derivative of distance with respect to time,
dv . d2s
a =
= s
dt " d*2
Deceleration is negative acceleration.
As with velocity, we distinguish between instantaneous
acceleration and average acceleration.
The third derivative of distance with respect to time is the rate
of change of acceleration, which, however, has no practical
application.

898
Chapter 30 MOTION
An object's motion along a straight line is registered from
time 0. At t seconds the position from a fixed point on the line
is
s(t) = (t2 - 2 t) meters.
Find: a: the acceleration;
b: the initial velocity and speed;
c: the velocity at t = 2.5 s;
d: the velocity when its directed distance from the
fixed point is 3 m.
With the velocity v, and acceleration a,
v(t) = s = 2t-2
a(t) = s = v = 2 .
v
s(t) = t2-2t
2 3 4
a
k
v(t) = 2t-2
T 1 1-
2 3 4
4
3
2
1
-1
a CO = 2
i i i
12 3
i '
4
a:
The acceleration is 2 m/s2.
b:
At time t = 0,
u(O) = 0-2 = -2.
The initial velocity is -2 m/s; the initial speed is 2 m/s.
The negative value of the velocity means that the object
initially moves in a direction opposite the one determined as
"intended", or positive.
c:
v (2.5) = 2-2.5-2 = 3 m/s .
Thus, at 2.5 seconds after start, the velocity is 3 m/s.
d:
Since s (t) = t2 - 2 t, we have
t2-2t -3 = 0
(t + 1) (t - 3) = 0 ; t = 3 s ; t = - 1, discarded.
v (3) = s (3) = 2 (3) - 2 = 4 m/s .
On completion of a distance of 3 meters, the velocity is 4 m/s.

Section 30.2 Differentiating Distance and Velocity
899
An experiment shows that from time t, a marble falls freely the
distance
s (t) = 4.9 t2 .
Determine when and at what velocity the marble strikes the
ground after being dropped from a window 44.1 meters above
street level.
v (t) = s = 9.8 t.
44.1 = 4.9t2 ; t = 3 s.
v (3) = 9.8-3 = 29.4 m/s.
Thus, the marble reaches the ground after 3 s at a velocity of
29.4 m/s.
An object is thrust upwards and reaches a height
s = (73.5 t - 4.9 t2) meters
at the end of t seconds.
Determine the highest point of the object's ascent.
v(t) = s = 73.5-9.8*.
The object reaches its highest point when v = 0;
0 = 73.5 - 9.81 ; t = 7.5 s.
Thus, the highest point reached is
s(7.5) = 73.5. 7.5-4.9 (7.5)2 = 275.625* 275.6m.
(Destination, Phase?
After a convivial evening in 'Edinburgh, an Englishman and a
Scotsman had embart^ed on the same train.
Said Jones: "It's a wonderful thing this new railway merger,
isn't it.9
Answered WladAlstair: "Aye, it is that, man. I'm gaun to
Aberdeen an' you're gaun to London an' we're baith on the same
train!"

900
Chapter 30 MOTION
30.3 Integrating Acceleration and Velocity
Since
we have
a = v and v = s,
v(t) = la ■ dt
s(t) = [v • dt.
In particular if a is constant, then
s(t) = J [a dt2 = \at +Ci dt = -^ at2 + Cit + C2.
A stone is dropped from the roof of a building and strikes the
ground after 4.5 seconds. Acceleration of gravity is 9.8 m/s2.
Determine the height of the building; disregard air resistance
of the stone.
v(t) = J 9.8 dt = 9.8 * + Ci;
at £ = 0, i> = 0soCi = 0andi>00 = 9.8t.
s(t) = \9.8t dt = 4.9t2 + C2;
at t = 0, s = 0, so C2 = 0 and s (t) = 4.912.
s (4.5) = 4.9 • 4.52 & 99 m, which is the desired height.
An aircraft requires 16 seconds and 768 m of airstrip to become
airborne. If we assume acceleration to be constant, what is the
takeoff velocity?
After roll-out to start, and with ground brakes engaged,
the aircraft races its engines to build up thrust; when
the brakes are released, the aircraft starts down the
runway with maximum acceleration.
v (t) = ja d t = a t + C\;
at t = 0, v = 0 and at = 0, so C\ = 0 ;
v(t) = at.
at2
s(t) =jv dt =jat dt =-z-+C2;
at
at t = 0, s = 0 and — = 0, so C2 = 0 ; s(t) =
At takeoff (t = 16 seconds),
atz
2 '
Hence,
_o a • 162 2-768 „ , 9
768=^—; a =-^- =6m/s2.
i; (16) = a • 16 = 6 • 16 = 96 m/s * 346 km/h.

901
30.4 Velocity Vectors and Acceleration Vectors
Velocity has direction as well as magnitude (speed) and
therefore is a vector quantity, and so is acceleration.
To find velocity and acceleration - and their components - for
objects moving in curvilinear paths, the use of vector analysis
is needed.
p. 608 The derivative of a vector a is a vector whose components are
the derivatives of the components of a. Thus, if the vector
displacement of a moving particle is
r (t) = xi +yj +zk,
where radius vector r is a vector from the origin to the particle
at time t in a three-dimensional orthogonal coordinate system,
then the velocity vector v is
p. 609 with the magnitude, or speed,
\v(t)\ =
^WW^W'
the acceleration vector a is
d^r d2x . dPy . d%
with the magnitude
A projectile is fired at an angle of elevation 0 and with an
initial speed of vq. If air resistance is disregarded, determine
the velocity vector v and position vector r at any time t.
Since the force of gravity acts in a downward direction, the
acceleration vector is
a = -gj.
v(*) =-jgi dt = - gt j + Ci,
where Ci is a constant vector;
aU = 0,v(0) = Ci,so
v(t) = v(0)-gt j.
Further integration gives the position vector
r(t) = j(v(0)-gt j) dt = tv(0)-±gt2j + C2;
r (0) = 0, so C2 = 0, and
(1) r(t) = tv(0)-^gt2j.

Chapter 30 MOTION
(vq sin 0)j
x
(vo cos 0) i
From the graph we have
v (0) = (vq cos 6) i + (vo sin 0) j
and can rewrite (1),
r CO = t [(v0 cos 0) i + (i;0 sin ¢) j] - j # £2 j
= £ (i>o cos 0) i + ( t vq sin 6 - — g t2 ) j . •
From r (t) of the above problem, we obtain the parametric
equations
X = Vq t COS 0
y = v0t sin 0-2 gt2
An object of mass m describes a circular path of radius r; its
angular velocity, the rate of change of the angle swept by the
radius vector in unit time, is a constant co .
For time t, find
a: the velocity v and the speed of the object;
b: the acceleration a and its magnitude;
c: the force vector F necessary to produce the motion.
Orienting the axes so that the center of the circle is at the origin
and the object is on the positive x-axis when t = 0, the polar
coordinate 6 of its position at time t must be co -1.
The path is thus described by the parametric form of the
equation for a circle of radius r centered at the origin of an
orthogonal coordinate system,
x = r cos co t 1
y = r sin cot J >
the position of the object is described by
r 00 = r (i cos co t + j sin cot).
a:
v (t) = r = co r [i (- sin co t) + j cos cot] .
speed = | v (t) \ = cor \ sin2 cot + cos2 cot = cor

Section 30.4 Velocity Vectors and Acceleration Vectors yuo
b:
a (t) = v = co2 r [(- cos cot)\- (sin co t) j]
= - co2 r (i cos cot +j sin cot)= - co2 r,
which shows that the acceleration vector has a direction
opposite to r and thus is directed towards the center of the circle.
| a(t) | = co2 r \ sin2 cot + cos2 cot = co2 r .
c:
F is the product of mass and vector acceleration,
F (t) = ma(t) = - mco2 r (i cos cot +j sin cot) .
The force vector is pointed towards the center of the circle
(centripetal force, meaning "center-seeking").

904
Chapter 30 MOTION
30.5 Space-Time; Mass and Energy
Special Theory of Relativity
EINSTEIN Albert
(1879 -1955)
General Theory of Relativity
Lorentz Transformations
LORENTZ Hendrik Antoon
(1853 -1928)
Contraction
of length
When Albert Einstein, in 1905, presented his special theory of
relativity, he had built on recent findings of the
electromagnetic nature of light and was able to break away from classic -
Newtonian - physics, where time is viewed independent of
space.
Einstein's special theory of relativity is called special because
it is concerned only with non-accelerated relative motion.
Einstein's general theory of relativity, introduced in 1915,
addresses, on the other hand, the problem of gravity and
accelerated motion, and indicates that gravitational forces
also affect electromagnetic radiation - which is energy
and thereby equivalent to mass. Among other things, the
general theory of relativity proposes that gravitational forces
bend rays of light, and this has been validated in astronomical
observations.
The special theory of relativity is based on two postulates:
1. In any two systems that move relative to one another with
constant velocity, all physical laws and principles are
expressed in the same mathematical form.
This implies that no experiment can be made to establish
whether the system of reference is at rest or moving with
constant velocity.
2. The speed of light in vacuum has the same constant value,
independent of the velocity of the source or the recipient.
Einstein abolished not only the thought of absolute space but
also that of absolute time, concepts that had been accepted but
never confirmed in Newtonian physics. Thus, Einstein
showed that there is no absolute simultaneity at different
locations; the present time - the "right now" - exists only at
the location of observation: now is only here. To understand
this, one must get used to viewing time as a coordinate that,
like the space coordinates, can be transformed.
Transformations of the coordinates of space and time take
place between two systems of reference moving at constant
velocity relative to each other; length and time, and also
mass, depend on the relative motion.
The effects of these space-time transformations, known as
Lorentz transformations after the Dutch physicist H. A.
Lorentz, are manifested in the following formulas. The
observer is located in a system of reference S that moves at a
constant velocity v relative to the system S' of the observed
object; the speed of light in vacuum is c.
1. A rod resting in system *S", where its length is Iq, has when
remeasured in system S shortened to

Section 30.5 Space-Time; Mass and Energy
905
Dilation
of time
Mass and
time
2. If time measured in system *S" is To, the observer in system
S finds the longer period of time T,
To
T =
V
(-
A consequence of the above two formulas and the theory of
relativity is that no velocity can be greater than the speed of
light in vacuum.
3. An object that, at rest in system *S", has the mass juq has in
system S the mass ra,
m =
m0
V
/^2
v<
Launched from planet ^-36 at the beginning of the planet's
calendar year 1200, a spaceship with a crew of 300 travels at
99.99% warp speed (99.99% of the speed of light) for exactly
3 years by the ship's clock, before returning to the planet. What
is the actual 4^36 calendar year of return?
V
* 212.1,
0.99992 c2
indicating 1412 as the year of return. •
Our spaceships attain a mere 0.005% of the speed of light; a
3-year space travel would make our astronauts age only about
12/100ths of a second less than the people remaining on Earth.
Time Warps
'"Why did you breat^Vessa's dotf?"
'"Because then she hit me."
i&srtssa
- '"What's the hurry, Mr. Schein?"
- "Wiy past is catching up with mel"

906
Chapter 30 MOTION
Einstein's Energy-Mass
Relation
SI multiples: p. 54
Einstein's energy-mass relation
E = m • c2
states that an object's total energy is the product of its mass and
the speed of light squared.
If an object with the mass mo is accelerated to the velocity v,
then it has been provided with the kinetic energy E^,
1
2£k = m • c2 - iuq • c2 = mo • c2
.V
- 1
1 -
r.
V
V
This relativistic expression for kinetic energy can for very
small values of vie be rewritten
mo • c2
V
- 1
1-
& mo • c2
V
1 +
r
V
Kc
- 1
& mo • c2
lfv
1 + 2{7
- 1
mo • u2
which is the classic formula for kinetic energy.
The energy radiated every second by the Sun is 360 • 1024 J and
the speed of light is 300 -106 m/s; find the Sun's loss of mass per
second. [J (joule), m (meter), s (second), and, for mass, kg
(kilogram) are coherent units of the International System of
Units (SI).]
The relation E = m • c2 means that every energy unit
corresponds to the mass m = Elc2, so the Sun's loss of mass per
second is
360 • 1024
m = 7T = 4 • 109 kg = 4 teragrams. •
(300 • 106)2
HEISENBERG Werner Karl
(1901 -1976)
Heisenberg's Uncertainty Principle; Measuring the TLtusive
The uncertainty principle, presented in 1927 by the German
physicist Werner Heisenberg, states that position and velocity
of a particle cannot, even in theory, be measured exactly at
the same time: the more accurately one knows the position, the
less accurately one can know the velocity, and vice versa. We
limit our account to the following case history:
There had been a series of nightly breakzins and thefts around
zvhere Angus MdTaggart had his small croft. One night he heard a
foreign noise from his pigsty, so he put on his breeches and went
out to have a tookzsee.
(When he came bacfi^ in, his ivife understandably wanted to fqww
if all the piggies were stitt there, a(t twenty of them.
Said Angus: "Och, lassie, Ahjuist dinna fen. !Aft sure counted
the nineteen o'them, but Ah spied a wee lit fella wha ran aboot sat
perfectly fast Ah couldna richtly charge ma mind to find him a
place tae count him.>}

907
Chapter
31
HARMONIC ANALYSIS
Page
31.0 Historical Notes 909
31.1 Fourier Series 910
31.2 Expanding Discontinuous Functions 915
31.3 Expanding Even or Odd Functions 918

908

909
31.0 Historical Notes
WIENER Norbert
(1894-1964)
American mathematician;
founder of the science of
cybernetics, the study of
mathematical structure of
control systems and
communication systems in
living organisms and in
machines. Wiener made
important contributions to
harmonic analysis, a field
that he widened extensively.
Tfie somewhat snobbish point of view of the purely abstract
mathematician would draw but little support from mathematical
history. On the other hand, whenever applied mathematics has
been merely a technical employment of methods already traditional
and jejune, it has been very poor applied mathematics, The
desideratum in mathematical as well as physical worf^ is an
attitude which is not indifferent to the extremely instructive
nature of actual physical situations, yet which is not dominated by
these to the dwarfing and paralyzing of its intellectual originality.
Viewed as a whole, the theory of harmonic analysis has a very fine
record of this sort. It is not a young theory, but neither is it yet in
its dotage, There is much more to be learned and much more to be
proved.
Norbert Wiener, 7ihe Historical Background of Harmonic Analysis (1938)
d'ALEMBERT Jean Le Rond
(1717-1783)
EULER Leonhard
(1707-1783)
BERNOULLI Daniel
(1700-1782)
FOURIER Jean Baptiste Joseph
(1768-1830)
In the 18th century, the French mathematician and physicist
d'Alembert and the Swiss mathematician Euler described
the vibration of strings by means of sums of arbitrary
functions, and Daniel Bernoulli, a member of the celebrated
Swiss family of mathematicians and scientists, used
trigonometric functions.
The use of trigonometric functions was further developed by
the French mathematician and physicist Joseph Fourier in his
Theorie analytique de la chaleur ("The Analytical Theory of
Heat"), published in 1822. Fourier showed that the conduction
of heat in solid bodies may be described by infinite series of
sines and cosines.
THEORIE
ISILTTIQUE
DE LA CHALEUR,
P** M. FOURIER.
A PARIS,
CHEZ millN DIDOT, PERE ET PILS,
i8aa.

910
Chapter 31 HARMONIC ANALYSIS
Fourier's work stimulated research in other areas, showing
that the type of series used by Fourier is a mathematical
prerequisite for the solution of periodic phenomena encountered
in practically every branch of science and technology, such as
the design of electrical circuits, the prediction of tides and sun-
spots, and the analysis of sound and electromagnetic waves.
The method of expressing periodic functions as sums of sines
and cosines is referred to as harmonic analysis.
31.1 Fourier Series
Dirichlet's Conditions
DIRICHLET Peter Gustav
(1805-1859) Lejeune
German mathematician
The principal idea of a Fourier series expansion is to represent
a function foi period 2% as an infinite series of trigonometric
functions, so that
fix) = -~- + a\ cos x + a<i cos 2x + ... + b\ sinx + b<i sin 2x ...
oo
fix) = -~- + 2^ ian cos nx + bn sin nx),
/i = l
where ao, an, bn are constants, n is a positive integer, and x
takes values between -oo and +©o. Writing the first term as —
instead of ao is, as we shall see, convenient for computation.
The notable quality of a Fourier series expansion is that it can
be used for functions that are represented by different
expressions in different parts of the interval.
The theory of Fourier series is complex and is still the object
of intense research. For the vast majority of mathematical
problems or physical applications the representation above is
valid for every x, because functions f arising in practice
ordinarily satisfy Dirichlet's conditions for convergence of
the Fourier series to f. Since sines and cosines both have
periods of 2n radians, the above expansion is unchanged
by replacing x with ix + 2k 7i), where k is an integer, which
corresponds to f having period 2 n.
Fourier Coefficients
While derivatives are used to find the coefficients of Taylor's
series, the coefficients of the Fourier series for f
oo
— + X, ian cos nx + bn sin nx),
/1 = 1
are determined by evaluating certain integrals which exist
provided that/*(x) is continuous in the interval —n < x < n.

Section 31.1 Fourier Series
911
ao To find an expression for the coefficient ao, we integrate f(x)
with respect to x from x = -n to x = n. We assume that f(x) is
the sum of the series and that the integral of the sum is the
sum of the integrals even though there are infinitely many
summands.
K
K
\f(x) dx =y J dx + £
f K K \
an cos nx dx + \bn sin nx dx
-K
- K
\-K
-n
J
oo
f r« + I
Z I — 7C /1 = 1
a
— sin nx - -^- cos nx
n n
n
L7C
/i = i
a
n
— [sin n n - sin(-rc n)]
bn.
n
[cos n 7i - cos(-rc k)]
oo
71-a0 + X
/i = l
/2 a
/i
26
V
^1
sin n n -
n
*-. 0
oo
7C • ao + X (0 - 0) = ao 7C.
/i = l
7C
ao
= I \f{x
) dx
-K
On To find an expression for an, n > 1, multiply f(x) by cos ra x,
where ra is a positive integer, and integrate with respect to x
from x = -ntox =n.
n
K
f(x) cos mx dx = — J cos rax dx
-7C ~n
n
oo -
y an cos 7ix • cos mx dx
/i = l J
-7C
7C
X fr/i sin nx • cos rax dx
/i = l J
-7C
which we integrate term by term:
7C
— J cos rax dx =
2ra
K
sin ra x = 0 (for all ra).
-7C
- 7C
To evaluate n
an cos nx • cos rax d x,
-n
distinguish two cases, m*n and ra = n .

912
Chapter 31 HARMONIC ANALYSIS
m*n
K
a
n
cos nx
• cos mx dx
-K
n
= -^- [cos (n - m) x + cos (n + m) x] dx
-n
a
n
sin (n - m)x sin (n + m)x
+ -
n - m
n + m
n
= 0
-*-n
m = n
n
an cos nx • cos mx
dx
-K
K
K
= an cos2 nx dx = -^- (l + cos 2 n x)
dx
-K
-K
= -r- x + -— sin 2nx =n - an .
2 |_TC 4n I _ ^
Thus, n
an cos nx
cos 772 x dx =
-n
0, ifn^m
K - CLn , if 71 = 772 .
7C
6^ sin nx • cos 772x dx
-K
n
b r
= "o*" [sin (71 + 772) x + sin (71 - 772) x] dx
-7C
'n
cos (71 + 772)x cos (n - m)x
—+-
71+772
72-772
7C
-<-7C
= 0 (for all 772 and n\ m^n) .
Thus,
7C
/Xx) cos 71 x dx =
-7C
0 if 772 ^ 71
K - an if 772 = 71
7C
1 r
aw = — /*(x) • cos Tix dx.
-K
bn To find an expression for bn, multiply f(x) by sin mx, where m
is a positive integer or zero, and integrate with respect to x
from x = -71 to x = 71.

Section 31.1 Fourier Series
913
n
K
f(x) sin rax dx = — sin rax dx
-K
-K
K
r
+ X \ an cos ax sin rax dx
-K
n
X bn sin rue sin /nx dx.
/i = l J
-7C
which we again integrate term by term:
K
a0 f j
— clq sin /72JC cbc = -
2ra
n
COS 772 71 = 0 .
- n
-n
n
n
an cos ax sin mx dx = -^- [sin (ti + ra) x + sin (a - ra) x]
dx
-n
-n
On
2
cos (a + ra)x cos (ti - ra)x
— +
a + ra
a — ra
n
-*-k
= 0 (for all ra and a; ra * a) .
To evaluate
7C
6^ sin a • jc sin rax dx,
-n
we again distinguish two cases, ra * a and ra = a .
7C
ra^a
bn sin ti • jc sin /n x
dx
-K
K
b r
= -^- [cos (ti - ra) x - cos (a + ra) x]
dx
-K
'n
sin (a - ra) x sin (a + ra) x
a - ra
a + ra
n
-1 -K
ra = a
= 0 (for all ra and a).
K
bn sin ax • sin rax dx
-K
K
K
= bn \ sin2 wa: (k = -^- | (^ ~ cos 2 m jc)
dx:
-7C
-7C
'/I
JC -
sin 'lax
la
-i 7C
= n-bn
J -7C

914
Chapter 31 HARMONIC ANALYSIS
Thus,
n
fix) sin nx dx =
0
if n * m
-n
71 • bn if n = m
n
1 r
bn = — fix) sin nx
dx
-71
Summing up: If a function/*(x) is continuous in -7i < x < n, it
has the series
-+^ (aw • cos nx + on • sin nx),
/i = l
where
n
1 r
ao = — J /*(*) dx ;
-n
K
an = — fix) cos nx dx ;
-7C
K
bn = — /Xx) sin nx dx .
-7C
There exist continuous f such that the Fourier series for f does
not converge to fix) for every x. If, however, f' is also
continuous, then we do have for every x that
oo
fix) = -5- + ^ (aw cos nx + 6^ sin nx)
/i = l
* * *
9¾^
A remarlqi6le lecture, (Professor.
(Iftan/^you. What impressed you?
your splendid integration with the topic and your
disintegration of the listeners.

915
31.2 Expanding Discontinuous Functions
When using power series expansion, such as Taylor series, we
are limited to continuous functions; with Fourier series, we
may also expand discontinuous functions.
Hitherto we assumed f(x) to be continuous in -n < x < n. If fix)
is discontinuous for x = xq in the interval -n < x < n and
if fixQ+) = lim fix) and f(xo_) = lim fix) exist,
x -* #0+ x —> x0-
then the coefficients can be determined as follows:
Xq
U
If If
an = — fix) dx + — fix) dx:
71 J 71 J
-n
Xq
Xq
U
an = — fix) • cos nx dx + — fix) • cos nx dx;
-n
Xq
Xq
K
bn = — fix) • sin nx dx + — /*(jc) • sin nx dx.
-7C
Xq
The Fourier series will converge to fix) at every point where
fix) is continuous and tor [/*(*+) +/ GO] at every point where
fix) has a jump of discontinuity.
Expand the function
f(x)= <
r 0; -7C<x<0
1; 0<x<7C
fix + 2kn)
k is an integer
-2%
1^
■o
-K I 7C
JC
27C
f is a periodic function with the period 2n and may be expanded
into a Fourier series.
Our principal concern is to find the coefficients of the Fourier
series «,
-+^ ian cos nx + bn sin nx) .

916
Chapter 31 HARMONIC ANALYSIS
We observe that all integrals over the interval [-71, 0] which
contain f(x) as a factor of the integral are equal to zero. Thus:
n
OQ
= Ijldx=l;
0
K
a
n
1 r
= — 1 • cos n x dx = 0 (ra = 1, 2, 3,...);
7C J
0
n
bn
= — 1 • sin nx
n J
cbe
(n = 1,2,3,...);
0
1_
n
-i r
— COS 71X
n Jo
= cos nx
n n L J
0
7C
0 for even n
2
71 7C
for odd n .
The expansion of f(x) contains a constant term and sine terms
only.
1 2 (sin x sin3x sin5x sin7x
\
fix) = 2 + nV 1
+ ...
J
if x^kn
As guaranteed by the theory, if x = k 7C, the series is
I + | (0 + 0+...) = | I/Xxj +/Xxj] .
Graphic Approximation
As shown by the following graphs, just a few terms of the
Fourier expansion may suffice to give a fair representation of
the graph of f(x):
2 simc
x

Section 31.2 Expanding Discontinuous Functions
917
1 2 /sin* sin3jc\ .
JC
1 2 /sin* sin3jc sin5jc\
X
-271 -71
7C 27C
1 2 /sin* sin3jc sin5jc sin 7x \ „ >.
2+7ln- + ^-+-5- + ^7^/ *fM
► JC
A completed, accurate graphic representation of the Fourier
expansion of f(x) requires an infinite number of terms and is
therefore unattainable in practice.

918
Chapter 31 HARMONIC ANALYSIS
31.3 Expanding Even or Odd Functions
cf. Even and Odd Functions:
pp. 340 and 508
A function f is even if f(-x) = fix) and odd if f(~x) = -f(x).
A cosine function is an even function; a sine function is odd.
The product of two functions that are both even or both odd is an
even function, while the product of one even and one odd
function is odd. For the evaluation of Fourier series, it will
save work if one can predict whether products of functions are
even or odd and realize that an area integral is considered
positive above the x-axis, negative below.
y = g(x)
x
y
j(^i[»[i[ *»'!* »•»*»*1
^¾¾¾¾¾¾¾¾¾^
-71
y = h(x)
7
... 1 »
III
Hi
$$-|ir
X
The function g(x) in the above graphs is an even function over
the interval x = -k to x = K and h(x) is odd:
71
n
n
-71
\g(x)dx = 2 tg(x)dx; [h(x) 6x = 0.
0
-7C
From this, two useful theorems emerge to simplify the
evaluation of Fourier series expansions of even or odd functions.
Even Functions
If f(x) is an even function defined over -n < x < 7C, then its
Fourier expansion reduces to a cosine series.
In the Fourier expansion for f,
oo
"o" + /u ^an cos nx + "n sin nx),
/1 = 1
we have, since /"is even,
7C
7C
Oq
= ljf(x)dx = ljf{x)
dx
n
0
The product of two even functions is even; the cosine function
is even:

Section 31.3 Expanding Even or Odd Functions 919
K K
If 2 r
an = — fix) • cos nx dx = — f(x) • cos nx dx.
-n 0
The sine function is odd; the product of an even function and
an odd function is odd:
K
1 r
bn = — f(x) • sin nx dx = 0 .
-K
Thus, for the even function fix), we have the Fourier series
oo
«0 v
TT + L an cos nx;
z /i = i
that is, Fourier expansion of even functions reduces to a cosine
series.
Odd Functions
If f(x) is an odd function defined over -n < x < n, then its
Fourier expansion reduces to a sine series.
Since fis odd, we have
K
1 r
clq = — f{x) dx = 0
-n
The cosine function is even; the product of an odd function
and an even function is odd:
K
1 r
an = — /*(jc) • cos nx dx = 0 .
-n
On the other hand, a sine function is odd; the product of two odd
functions is even; therefore,
K K
lr 1 r
bn = — /*(jc) • sin nx dx = 2 — /*(jc) • sin nx dx.
-n 0
Thus, for the odd function fix), we have the Fourier series
oo
X bn sin nx;
/i = l
that is, Fourier expansion of odd functions reduces to a sine
series.

920
Chapter 31 HARMONIC ANALYSIS
Expand the triangular function
f(x)= {
\x\; -n<x <n
fix + 2kn)
k is an integer
x
f is a periodic function of period 2 n
Since f is even, bn = 0.
oo
fix) = — + X a/l cos nx\
A /1 = 1
7C
OQ
=kJ>i
(be =
2 i* x2
71
0
0
T = n;
7C
an = — \ \x | cos ^1 * dx.
0
Integration by parts gives
«/i =
2
7C
x sin fix cos nx
+
-, n
n
ni
0
r
2 in
= 0 + cos nx = <
7C m^ In
0 for even n
4
7C IV
for odd ^1 .
Numerical Value of n
fix) =
Hence,
n 4
n
cos* cos3x cos5jc cos7jc
—:— + — + — + — +
32
52
72
for all x.
Incidentally, setting x = 0, the series may be used to find a
numerical value of n; thus,
n 4 fcosx cos3x cos5jc cos7jc
0 = — - — —:— + — + -— + -— + ...
2 n I 1 32 52 72
or
7C2 =
K 4 (^ 1 1 1
2 n I 32 52 72
111
811 + ^i+ ^+ ^2+-

Section 31.3 Expanding Even or Odd Functions
921
Expand the sawtooth function
fix) = J
x ; -71 < X < 71
fix + 2kn)
k is an integer
x
f is a periodic function of period 2n.
Since /*is odd, an = 0 and aw = 0; ra = 1, 2, ... .
7C
/X*) = X fr/i sinrcje; bn = — j fix) sin rue dx
/i = i n
0
Integration by parts gives
K-2-
n K
-x cos nx sin rue
+ T
->n
n
n4
0
where the second term is zero, and so
bn =
2 cos nn
n
Hence,
oo
oo
fw = X (^)(-1^
-1
sin nx
sin nx
n = l
.2 1(-1,.-. 2L
n = l
= 2
sinx sin2x sin3x . .. ,^,
+ — ... , if x * (2 k + 1) n
v 1 2 3
Once again, the series converges to the average
0=h\f(x+)+f(x_)]
if x = i2k + l)n.

922
MMMtamtnd«
**"
From F. Gafurius, Theorica Musice, Milan, 1492.
[D. E. Smith, History of Mathematics; Dover, 1958.]
Music, and its theory, was for the Pythagoreans (6th to 4th
century B.C.) one of the four mathematical categories of
science: arithmetic, geometry, music, astronomy.
PYTHAGORAS Legend supports that Pythagoras of Samos discovered that for
(c. 580 - 496 B.C.) strings under equal tension, the lengths should be 2 to 1 for the
octave of a note, 3 to 2 for the fifth, and 4 to 3 for the fourth.
From the knowledge that whole numbers determine musical
intervals, the Pythagoreans inferred that the intervals
between the heavenly bodies were governed by the laws of
musical harmony, and that the same harmony is the true ruler
of everything in nature.
From Georg Reisch, Margarita phylosophica (first edition, 1496; here
reproduced from a 1583 reprint).
Music had a prominent position in schools of the Middle Ages
and the Renaissance. Music was one of the seven liberal arts:
geometry, astronomy, arithmetic, music, grammar, dialectics
(logic), and rhetoric.

923
Chapter
32
METHODS OF APPROXIMATION
Page
*** 4.7 Reliability of Digits and Calculations 161
32.1 Negligible Terms 925
32.2 Interpolation 926
32.3 Graphic and Iterative Methods 931
32.4 Numerical Integration 947
* Indicates cross-reference

924

925
32.1 Negligible Terms
A quantity
where
A = (a0 + Aa)P = atf>(l + e)P,
A a
a0
= e « 1,
may be calculated with acceptable accuracy by observing that
(l + e)P = l+p-£+ p{p~X) .£? + ...,
where £2 and higher terms are negligible.
V 100.5 = 10
f. 0.5^
1 +
V
100
= 10 (1 + 0.005)m * 10 (1 + 0.0025)
Vl22 = (125-3)1^ = 5
f
\
3 ^3
125
= 10.025
* 5(1-0.008) = 4.960
Express 9.9734 9 in scientific notation.
9.97349 = 109 (1-0.0266)9 * 108 (1 - 0.2394) = 9.76 • 108
A circular cylindrical container has a nominal diameter
d - 5 length units and a nominal height /& = 10 length units; on
measurement, we find d = 5.015 h = 9.960 length units. How do
these deviations affect the bottom area and the total volume?
d = 2r = 5.015 = 5(1 + 0.003) h = 9.960 = 10(1-0.004)
A = 7ir2*Ao-(l + 2- 0.003) * 1.006 • Aq
V = nr2h *V0'(1 + 0.006) (1 - 0.004) * 1.002 • V0
* * *
M <HegCigibk
IQTester: If I'm 1.78 meters tall, weigh 74 /qtbgrams, commute
97 kilometers a day, and twice the square root of my
telephone number is 2431, what is my age?
Testee: 48, sir.
/Q, Tester: flight! (But how did you figure it out?
Testee: I've a cousin who is 24 and he's only half nuts.

Chapter 32 METHODS OF APPROXIMATION
32.2 Interpolation
When the values of a mathematical function corresponding
to two arguments (independent variables), called basic points,
are known, the value that corresponds to an argument
intermediate to the basic points can be found by interpolation.
The simplest form of interpolation is linear interpolation,
which presupposes that the variation of the function value can
be described by a straight line passing through the two basic
points.
► x
X
1.003
1.004
1.005
1.006
1.007
1.008
• • •
5.03
5.04
5.05
5.06
5.07
5.08
lg*
0.001 30
0.001 73
0.002 17
0.002 60
0.003 03
0.003 46
0.701 57
0.702 43
0.703 29
0.704 15
0.705 00
0.705 86
A \gx
0.000 43
0.000 44
0.000 43
0.000 43
0.000 43
• • •
0.000 86
0.000 86
0.000 86
0.000 85
0.000 86
If the arguments of the two basic points are x\ and x<i, and the
corresponding function values y\ and y<i, the value y of the
function for an argument x will be
yi +
X - X\
X2 ~X\
(y2-yi)
Referring to a table of common logarithms, we find that the
assumption of linearity of variation is permissible, and we
may apply the formula to calculate lg 1.0057 and lg 5.057:
1.005 7-1.005 ,,
lg 1.0057 = lg 1.005 + . (lg 1.006 - lg 1.005)
1.006-1.005
= 0.002 17 + 0.7 (0.002 60 - 0.002 17)
= 0.002 17 + 0.000 301 = 0.002 47(1) .
lg 5.057 = lg 5.05 + 0.7 • (lg 5.06 - lg 5.05)
= 0.703 29 + 0.7 • 0.000 86
= 0.703 29 + 0.000 602 = 0.703 89(2) .

Section 32.2 Interpolation
927
x eP° A e*
7.1 1212.0
7.2 1339.4
7.3 1480.3
7.4 1636.0
7.5 1808.0
7.6 1998.2
7.7 2208.3
127.4
140.9
155.7
172.0
190.2
210.1
232.3
2400
2200
2000
1800
1600
1400
1200
When a table shows that the function does not satisfy the
condition of linearity of variation, we may resort to graphical
interpolation by plotting the function values in a graph and
trying to draw a curve passing through the basic points.
For a demonstration, we choose the exponential function e*.
The value e7-5 may be calculated by linear interpolation as the
arithmetic mean of two adjacent exponential functions:
J.5
* <
1
2
1
2
± (e7.4 + e7.6)
- (e7.3 + e7.7)
1
2
± (e7.2 + e7.8)
= 2 (1636.0 + 1998.2) = 1817.1
= 2 (1480.3 + 2208.3) = 1844.3
2 (1339.4 + 2440.6) = 1890.0
v^
The results obtained differ from the table value of 1808.0 by 9.1,
36.3, and 82.0, respectively, corresponding to errors of 0.5%,
2.0%, and 4.5%.

Chapter 32 METHODS OF APPROXIMATION
X
75°
76°
77°
78°
79°
80°
81°
82°
83°
84°
85°
86°
87°
88°
89°
tan x
3.732
4.011
4.331
4.705
5.145
5.671
6.314
7.115
8.144
9.534
11.53
14.30
19.08
28.64
57.29
lg tan x
0.5719
0.6032
0.6366
0.6725
0.7113
0.7537
0.8003
0.8522
0.9109
0.9784
1.0580
1.1554
1.2806
1.4569
1.7581
Linear interpolation will generally yield a satisfactory
interpolated value for all normal monotonic functions provided
that the basic points straddling the sought value are chosen so
as to be as close to each other as known values permit.
Graphic interpolation may be a suitable method for functions
that rise very steeply with moderate changes of the argument,
e.g., the trigonometric tangent function above 80 degrees of
angle.
Linear interpolation is generally acceptable, however, as
shown by the graph of the tangent function and its logarithm.
The use of the logarithm of a function is often a possible
solution for interpolation, followed by conversion of the value
found to its antilogarithm.
35 sr
76 78
80
82 84 86 88
90
For greater accuracy of interpolation, several methods of
replacing the function with polynomial expressions that
approximate it to any desired degree of accuracy have been
suggested by Lagrange, Newton, Gauss, and other
mathematicians of renown.
All mathematical interpolation procedures consist of
replacing the original function with simpler functions which
provide exact agreement at the known basic points and
optimum approximation to the original function in the
neighborhood of these points.

Section 32.2 Interpolation
929
RENARD Charles
(1847 -1905)
Preferred Numbers
A special form of interpolation deals with the dividing up
of decade intervals of numbers into sub-intervals of
logarithmically equal width, so that consecutive points of division
form a geometric series with the common ratio
q= VlO,
where N may assume the values 5, 10, 20, 40, and, in
exceptional cases, 80, according to International Standard (ISO 3).
In the five series, prefixed by the letter R - after their inventor,
the French military officer Charles Renard -
R5
R10
5/— 10/—
step ratio V10 V10
approx.
value 1.58 1.26
R20
20,—
Vio
1.12
R40
R80
40/— 80/—
Vio Vio
1.06
1.03
preferred numbers are calculated to five digits and rounded to
three digits, as shown here for R 5, R 10, and R 20 (for tables of
preferred number series R 40 and R 80, see ISO 3).
Mantissas
000
050
100
150
200
250
300
350
400
450
500
550
600
650
700
750
800
850
900
950
000
Calculated Values
1.0000
1.1220
1.2589
1.4125
1.5849
1.7783
1.9953
2.2387
2.5119
2.8184
3.1623
3.5481
3.9811
4.4668
5.0119
5.6234
6.3096
7.0795
7.9433
8.9125
10.0000
Standard Series
R5
1.00
1.60
2.50
4.00
6.30
10.00
R10
1.00
1.25
1.60
2.00
2.50
3.15
4.00
5.00
6.30
8.00
10.00
R20
1.00
1.12
1.25
1.40
1.60
1.80
2.00
2.24
2.50
2.80
3.15
3.55
4.00
4.50
5.00
5.60
6.30
7.10
8.00
9.00
10.00
From the basic tables, further tables of preferred numbers may
be created by selecting every second, third, fourth ... n-th term
of a series.

930 Chapter 32 METHODS OF APPROXIMATION
The use of preferred numbers benefits industry by restricting
the number of machine element sizes, rotational speeds,
mechanical and hydraulic pressures, etc., and still satisfying
practical industrial requirements with a minimum of
components and reduced costs of production and stockkeeping.
In the electronics industry, resistors and capacitors are
graded according to another system of preferred numbers
based on the 12th root of 10, and rounded off to two figures,
10 -12 -15 -18 - 22 - 26 - 32 - 38 - 47 - 56 - 68 - 82 -100,
sometimes with the addition of 62, 75, and 91 from the 24th-root
series.
With interpolation perforce,
Jacl^plotted a much better course,
"With JUL up the hill,
Oh my! 9\[ptaspill,
Skateboarding down from the old water source!

931
32.3 Graphic and Iterative Methods
Real numbers of the independent variable (the argument) for
Real Zeros which the function is 0 are referred to as the real zeros of the
function. Thus, if f is a function, then a real root of f(x) = 0 is
a real zero of f.
p. 304 By the factor theorem, every real zero of a polynomial function
f corresponds to a first-degree factor of f(x). Consequently, a
polynomial function cannot have more real zeros than its
degree; e.g., a third-degree polynomial function has at the
most three real zeros, a fourth-degree at the most four.
First- and second-degree equations and occasional higher-
degree equations can always be solved by using algebraic
methods; formulas also exist for solving third- and fourth-
degree equations, but they are very complicated. If f is a
polynomial of degree five or higher, there is, generally, no
algebraic solution. Furthermore, there is no formula by which
we can find the exact roots of cos x = x or other mixed
trigonometric equations.
We realize that the number of equations that cannot be solved
by algebraic methods many times exceeds the number of
equations that are suitable for such methods.
When algebraic methods are not suitable, we can obtain
approximative real-number solutions by the use of graphic
methods, often in combination with iterative computational
methods by which a result is obtained by replication of a series
of operations that successively improves an approximate
result.
Graphs
Equations in One Variable
We may solve equations in one variable for real-number roots
if we change the standard form, clq xn + a\ xn ~1 + ... + an = 0, to
clq xn + apcn ~1 + ... + an = y
and graphically determine what points on the curve have y = 0.
Graphical solutions are usually time consuming. If the roots
are packed closely together, it can be virtually impossible to
locate the roots graphically.
• Find the roots of x3 - 3x - 1 = 0.
Write
x^ - 3 x - 1 = y
and use convenient values of x to find corresponding values
ofy:
X
y
-2
-3
-1.5
0.125
-1
1
-0.5
0.375
0
-1
0.5
-2.375
1
-3
1.5
-2.125
2
1

932
Chapter 32 METHODS OF APPROXIMATION
Plot the values of x and y on an orthogonal coordinate system
and sketch the curve of x3 - 3x - 1 = y :
*3-3x-l=:y
pp. 522 - 23
The graph of x3 - Sx - 1 = y intersects the x-axis in three
places; a third-degree equation has three roots and,
consequently, all three roots of the equation have real-number
values.
The equation
2 cos x - sinx = -2; 0 < x < 15
was solved by algebraic methods in Chapter 13. It may be
solved graphically by determining the abscissas of the points
of intersection of the curves of
y = 2 cos x ; y = sin x - 2 .
y = 2 cos x - sin x + 2
14.78
•*- x
y\ = 2 cos x
y2 = smx

Section 32.3 Graphic and Iterative Methods
933
Systems of Equations
2x+y = 5
x2+ y2 = 25
y
x2+y2 = 25
The graphs of the equations intersect at (0, 5) and (4, -3),
leading to the solutions
xi = 0
yi = 5
4x2+y2 = 201
x2 = 2y
I *2 = 4 ]
1 . y2 = -3 J
>
(-2, 2)
jc2 = 2y
4jc2 + y2 = 20
The graphs of the equations intersect at (2, 2) and (-2, 2),
leading to the real-number solutions
xi = 2
yi = 2]
1 *2 = "2
1 . y2 = 2
but the algebraic method gives additional solutions, whose
number pairs contain complex-number components,
x3 = V-20
y3 = -10
x4 = -V-20
y4 = -10

934
Chapter 32 METHODS OF APPROXIMATION
x2 = 2y
x - y = 1
By algebraic means, we find two complex solutions,
2 + V-^4 .
xi = = l + i
>v
yi =
V^4
= l
X2 =
J2 =
2-V^4
>v
= 1-i
-V-4
but there are no real-number solutions; since there are no
real-number solutions, the graphs do not intersect:
x-y = 1
= 2y
Graphic solutions are approximate, but by determining an
increasing number of x, y-points around an intersection of the
graph and the x-axis, the accuracy may be improved to any
required degree.

Section 32.3 Graphic and Iterative Methods
935
Iterative Methods
A real zero of a function f is the same as an x-intercept of the
graph of f There are several numerical procedures for
finding real zeros of functions; the two most frequently used
methods are the bisection method and Newton's method.
Bisection Method
p. 551
Assume that /*is a continuous function and suppose that f has
the real zeros a and b:
*»x
To find a by the bisection method, use initial values that
straddle a, say, x = %i and x = xz, giving f(xi) and f(xz)
opposite signs. Letting the midpoint of the interval [xly xz] be
x mv we have
x^ i xz
^Ml =—5—
x
To determine whether the zero of the function is in [xi, x^J or
in [xmv %z], we ask:
(1) Is
f(xi) > 0
and
fix Ml) < 0
If the answer to question (1) is no, the resulting question is:
(2) Is
f(xi) < 0
and
fix Ml) > 0
If either question (1) or question (2) has an affirmative
answer, then the x-intercept of f is in [x1? xmJ- On the other
hand, if neither question has an affirmative answer, then the
x-intercept of fis in [xmv xz\.

936
Chapter 32 METHODS OF APPROXIMATION
For our function f the x intercept is in [xi, xjmJ', zooming in
on [x\, xz]:
•*1"*m2
The interval [x\, xj^y\ is now bisected, yielding the intervals
[xi, xm2] an^ [xm2> %!•
We now ask:
(3)
Is
f(xi) > 0
and
fix M2) < 0
Since the the answer to question (3) is no, the ensuing question
is:
(4) Is
f(xi) < 0
and
f(xM2)>0
Question (4) is also answered no and, therefore, the x-intercept
must be in [xm2> xM^\ :
The bisection process is repeated until desired accuracy is
obtained.
Finding Initial Values
In order to obtain useful initial values, it is often necessary to
make a reasonably detailed graph of the function or, for a
function composed of different kinds of elementary functions,
to sketch the individual elementary functions.
If given the function fix) = lnx - sinx for 0 < x < 4, we could
after some rather lengthy calculations obtain its graph,
showing a real zero in the interval [2, 2.5] :
fix)
A
fix) = In x - sin x
x '

Section 32.3 Graphic and Iterative Methods
937
Since f(x) = \nx - sinx = 0, then lnx = sinx. The functions
yi = lnx and y<i = sinx are more easily sketched than f\ the
point of intersection of y\ and y<i is in the interval [2, 2.5],
coinciding with the x-intercept of the graph of f:
y1=]nx
x
Solve the equation
LUb JL —~ JL .
round off the result to two decimals.
Let f(x) = cos x-x; sketch y\ = cos x and y<i = x
y2=x
x
y\ - cos x
The graphs show a point of intersection between y\ andy2- The
size chosen for the interval in which the intersection is located
depends on the accuracy of the drawing; we choose the initial
values 0.6 and 1.
The midpoint M\ of the interval [0.6, 1] is
1 + 0.6
Mi =
= 0.8
Examine the intervals
[0.6, 0.8] ; [0.8,1] .
(1) Is
/X0.6) > 0
and
/(0.8) < 0

938
Chapter 32 METHODS OF APPROXIMATION
Since cos 0.6 - 0.6 > 0 and cos 0.8 - 0.8 < 0, the answer to (1) is
yes. Consequently, the real zero of f is located in [0.6, 0.8],
whose midpoint M<i is
0.6 + 0.8 nn
M2 = - = 0.7.
Examine the intervals
[0.6, 0.7] ; [0.7, 0.8] .
(2) Is
/X0.6) > 0
and
/X0.7) < 0
Since cos 0.7 - 0.7 <t 0, the answer to (2) is no. Therefore, the
next question is:
(3)
Is
/X0.6) < 0
and
/X0.7) > 0
Since cos 0.6 - 0.6 <t 0, the answer to (3) is no. Consequently,
the real zero of /"is located in [0.7, 0.8].
Further iterations of the algorithm produce the intervals
[0.7, 0.75],
[0.725, 0.75],
[0.7375 , 0.75],
and
[0.7375 , 0.74375] .
Hence, rounded off to two decimals, the sought root is 0.74 .
i

Section 32.3 Graphic and Iterative Methods
939
Real Zeros Unsuitable for the Bisection Method
The bisection method requires an x-intercept of the function
and therefore cannot be employed to find a real zero in an
interval where the x-axis is the tangent to the curve. For
example, consider
fix) = 2 x3 - 3 x2 - 36 x + 81,
whose graph for -5 < x < 5 is:
fix) = 2x3 - 3x2 - 36x + 81
x
p. 941
The bisection method is suitable for finding the real zero of f in
the interval [-5, -4], but not in [2, 4]; the latter interval is
accessible to examination by Newton's method.

940
Chapter 32 METHODS OF APPROXIMATION
Flow Chart for the Bisection Method
Iterative methods are suited for computer programming. In
the chart below, A and B are initial values, E is the error
tolerance:
/ ; /
midpoint = A±J.
<
A = A
B = midpoint
root at
midpoint
* In order to express the final estimate as a value
with an associated error tolerance, one must round
or truncate. In either case, using E as the cut-off
point in one's calculation rather than E/2, one can
easily end up with an estimate + tolerance (E)
which does not include the actual value.
With a computer program, one is less dependent on details of
the graph of the function and may without excessive loss of
time use initial values that are far apart.
There is always the possibility that the computer program
will place the demand on error tolerance too high, causing
inordinately long computations or even an infinite loop. It is
therefore important to specify not only the error tolerance but
also the maximum number of iterations; a real zero to the
specified accuracy, or the specified maximum number of
iterations (whichever occurs first) will stop the calculations.

Section 32.3 Graphic and Iterative Methods
941
Newton's Method
Seed
Newton's method requires a seed, that is, an initial rough
estimate or guess of the value of the root.
With a suitable seed, the tangent of the curve at the point
corresponding to the seed (xo) will intercept the x-axis. The
x-intercept may be located at a point that is at a shorter distance
or, as in the graph below, at an intercept {x{) farther away from
the value of the sought root.
y
A
seed at
y = f(x)
x
Using the tangent intercept at x\ as a new seed, we continue the
approximation and obtain a new intercept at x2. In this
manner, starting with the approximation x = xq, we obtain a
sequence xi, x2, x% ..., where successive values come ever
closer to the sought root.
y
A
seed at
y = fix)
x
The formula describing the steps of this iterative process is
f(*n)
xn + l - xn ft(x^ ,
wheren = 0, 1, 2, ... and /and its derivative are continuous.

942
Chapter 32 METHODS OF APPROXIMATION
Why?
p. 553
The slope of the graph of a differentiable function f has, at any
point [x, f(x)], the value f'(x).
The point-slope equation of a tangent line at point [xq , f(xo)] is
y -/X*o) = I/' (*o)] (x-xq) .
If this tangent line has the jc-intercept x\, we have the point-
slope equation
0-f(x0) = [f (*o)l (*i-*o)
and can express jci in terms of x$\ thus,
_ f(xp)
XI - XQ fl (jco) .
Repeating this process for the tangent line at [x\ , f{x\)\ and
expressing #2 in terms of x\, we find
and for ^1 = 0, 1,2, ..., the general formula
%n + 1 — %n
f'(Xn) '
Perils and Fallacies of Newton's Method
Unless one has a rather detailed understanding of the
function, Newton's method may give deceptive results;
generally, the bisection method is more reliable.
Newton's method may fail if one chooses a seed far away from
the sought root, leading to increasingly worse approximations
or to convergence towards the wrong root.
y = /W
x
If, in the course of approximations, a tangent line falls on an
extremum, then /"' (xn) = 0 and, since division by 0 is not
possible, xn + i will be undefined. Another choice of seed
usually takes care of the problem.
In some instances, the shape of the curve is such that Newton's
method will fail for any choice of seed other than the actual
jc-intercept. It is often helpful to do a few iterations of the
bisection method and then switch to Newton's method.

Section 32.3 Graphic and Iterative Methods 943
Determine V2 to five decimal places by Newton's method.
To determine y2 is equivalent to finding the positive root of
the equation
x2-2 = 0.
With f(x) = x2-2,
f' (x) = 2 x
and
_ _ xn ~ ^
%n + 1 — %n ~ ~
Z Xn
With xq = 1 as the seed,
1-2
x\ = 1 - —-— = 1.5 .
The iterative process is carried on until successive
approximations agree to five decimal places:
1.52-2
xn = 1.5- 0 ,., -. & 1.416 667
z 2 (1.5)
1.416 6672 - 2
x3 = 1.416 667- 2(1416667) 0 1.414 216
x4 = 1.414 216 - 2 (L4U 216) * 1.414 213
Hence, correct to five decimal places,
V2 = 1.414 21 .
• With an error tolerance < 0.0001, find the root of
x3 - x = -1
in the interval [-2,-1] .
Alternative I: Newton's Method
With f(x) = xs - x + 1,
fl(x) = 3x2-l
and
xn + 1 = xn ~~
%n %n "*" ■*■
o xn —1
With (-1.5) as a seed, xq = -1.5 and
^=-1.5-^^-^-5^1 =-1-34782 ■■■■
3 (-1.5)2 - 1
Discontinue the calculations when the difference between two
successive values of x is < 0.00005:
x1= -1.34782...
x2= -1.32520...
x3= -1.32471...
x4= -1.32471....

944
Chapter 32 METHODS OF APPROXIMATION
Hence, the sought root is -1.3247 ± 0.0001
y = f(x) = X3-X + 1
X
Alternative II: Bisection Method
We choose the initial values - 2 and -1.
The midpoint Mi of the interval [-2, -1] is
-2 + (-1) 3
"1 - 2
Successive calculations give
Mi
M2
M3
M4
M5
M6
M7
M8
M9
M10
Mn
M12
Mi3
MU
Mis
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
2 '
i
-1.5
-1.25
- 1.375
-1.312 5
-1.343 75
-1.328 125
-1.320 312 5
-1.324 218 75
-1.326 171875
- 1.325 195 312 5
- 1.324 707 03125
- 1.324 951171 875
- 1.324 829 101 562 5
- 1.324 768 066 406 25
- 1.324 737 548 828 125
The sought root is -1.3247 ± 0.0001. •
When Newton's method fails, the bisection method will
usually provide a value of the real zero; similarly, when the
bisection method fails, recourse to Newton's method is usually
profitable.
With an error tolerance of < 0.01, find the solution of
5
2x2
X
3--l —
27
in the interval [1,2] .

Section 32.3 Graphic and Iterative Methods
945
In the given interval, f shows a maximum on the x-axis;
consequently, use of the bisection method would fail to determine
the sought root.
x
y=f(x) = 2x2-x3-l
27
Newton's method takes care of the problem.
32
With fix) = 2x2-x3- —
fix) = 4x-3x2
and
z Xn'
X
n
%n + 1 ~~ %i
32
27
n
With 1 as a seed, xq = 1 and
X\ — 1
4 Xn O Xn
2-1-22
z x 27 32
4-3 " 27 '
Discontinue the successive calculations when the difference
between two successive values of x is < 0.005:
xi = 1.18518.
x2= 1.262 34.
x3= 1.298 50.
x4= 1.316 07.
x5= 1.324 74.
x6= 1.329 04....
Hence, the sought root is x = 1.33 ± 0.01 .

946
Chapter 32 METHODS OF APPROXIMATION
The best part of my new
math program is the
Pooh book that came
with it.
Flow Chart for Newton's Method
As with the bisection method, Newton's method is suited for
computer programming. In the flow chart below, the seed is xq
and the error tolerance E.
p. 940 *-note
"^r>*3^
"9low do you do 9{pthing?" askedTooh, after he had wondered for a
long time.
'"Well, it is when people call out at you just as you're going off to do
it, 'What are you going to do, Christopher %pbin, and you say, Oh,
nothing, and then you go and do it."
"Oh, I see," saidTooh.
"This is a nothing sort of thing that we're doing now."
"Oh, I see," said (Pooh again.
"It means just going along, listening to all the things you can't hear,
and not bothering."
"Oh!" said Tooh.
cIhey walled on, thinking oflfiis andrLhat, and by-and-by they came
to an enchanted place on the very top of the Jorest called Qalleons Lap,
which is si?(ty-something trees in a circle; and Christopher %pbin knew
that it was enchanted because nobody had ever peen able to count whether
it was si^ty-three or si?(ty-four, not even when he tied a piece of string
round each tree after he had counted it.
A. A. Milne, 7ihe House at Pooh Corner (1928). Illustrated by Ernest H. Shepard.

947
32.4 Numerical Integration
To evaluate definite integrals we have, up to this point, used
methods requiring that we first find the indefinite integral of
the given integrand; by such methods we have obtained exact
values.
Numerical integration, on the other hand, is the process of
finding an approximate value of a definite integral without
carrying out the process of evaluating the indefinite integral.
Approximate Integration The term approximate integration is used synonymously with
numerical integration.
The use of numerical integration to evaluate a definite
integral is required
o when the indefinite integral cannot be expressed in terms
of an elementary function;
o when the integrand is available only as a table of
numerical values or a graph, which is often the case in
scientific observations.
Numerical integration is possible only when the limit values
of the integral are known. It is often employed to evaluate a
complicated integral even if the indefinite integral can be
expressed in terms of elementary functions.
Although numerical integration cannot yield an exhaustively
defined value, it can be carried on to as many decimal places
as desired or as can be achieved in the allotted time.
This does not mean that the method of finding the indefinite
integral expressed in terms of elementary functions is
superfluous. On the contrary, it is often important to learn what
functions constitute the integral.
Numerical integration employs
o series expansion, or
o geometrically oriented methods.
Formulas of numerical integration are often referred to as
Quadrature Formulas quadrature formulas, recalling historic attempts to find
squares equal to surfaces defined by curved lines.
Infinite Series Expansion
Historically, the investigation of infinite series developed in
connection with problems of finding integrals.
If the limits of integration fall within the interval of
convergence for power expansion of the integrand, then an
approximate value of a definite integral may be obtained by
summation of the definite integrals of the early terms of the
series.

948
Chapter 32 METHODS OF APPROXIMATION
Find
sin x
x
dx
o
p. 771
Use the Maclaurin expansion of sin x, divide each term by x,
and integrate term by term,
sin x
x
_ x° x
dx = x- -—— +
x
sin x
x
dx =
3-3! 5-5! 7 • 7!
1 1
1- _ _. +
+ .
3-3! 5 5! 7•7!
0
-2lt
/W =
smx
2tc
4k
8n
By including an increasing number of terms in the sum, we
achieve ever-greater accuracy. Since the expansion produces
an alternating series, the error is bounded by the last term
used in the expansion.
Find [ e"*2 dx .
j
e~x dx = x-
JC3 JC5
31! 5-2!
y
I
f(x)=e-x'
Since e~x dx = 2 I e~x dx, we have
i o
fe-*2dx = 2 (l- T^— + X
31! 5-2!
As in the previous example, the error is bounded by the last
term used in the expansion.

Section 32.4 Numerical Integration
949
The Trapezoidal Rule
y
A
y=/W<
X
The simplest approximation of a curve y = f(x) extending
between two points P and Q is a chord connecting the two
points.
Letting the domain off be [a, 6], the area Ay between the chord
and the x-axis is a trapezoid with the length (6 - a) and the
parallel sides f(a) and f(b); thus
AT = \ (b - a) \f(a) + f(b)] .
The smaller the intervals that are replaced by chords, the better
the approximation of the area will be. In the graph below, [a, 6]
is divided into n sub-intervals of equal length, :
n
y=m
-. _ *
x1 X2 #3 xn_2 xn_1 xn
X
By adding the areas of the sub-intervals, we have
ATl + AT2 + ... + ATn = \-^r- lf<a> + /<*!>] +\^T- lft*l) + f^2)\ +...
n
n
1 b -a r l b -a , n
If (** -2) + fe - i)l + - —r— [ft** - l) + f(b)]
+ 2~
^1
b - a
= -27T \f(*) + 2Axi) + 2/(^2) + -. + 2/*(x„ _ i) + /(6)],
and the trapezoidal rule,
b
jf(x) dx * -^- \f(a) + 2f(Xl) + 2f(x2) + ... + 2f(xn _ i) + /(6)] ,
where n is a positive integer.

950
Chapter 32 METHODS OF APPROXIMATION
Approximate the values of the integral
l
j
o
rx dx
by the trapezoidal rule for n = 1, n
answers after the fifth decimal.
= 2, n = 4. Truncate the
71 = 1
y
A
_*-*2
1/2-^^^B
1/4
1/4 1/2 3/4 1
x
\e~x dx & ——7- (e" ° + e" 1)
0
2- 1
* 2
1+-1^ 0.68393....
V
71 = 2
y
A
3/4-1
1/2-1
-*-*2
1/4-^^bp
1/4 1/2 3/4 1
X
\ e~*2 dx * -^-£ (e-° + 2 e"1/4 + e"1)
0
2-2
*4
1 + 2
1 A
e0.25y
+ —
e
* 0.73137....

Section 32.4 Numerical Integration
951
71 = 4
_.-*2
X
1/4 1/2 3/4
i
0
e-*2d* ^ ^^(e-O+^e-^+^e-^+^e-^ + e-1)
2-4
* 8
/ 1
( 1
/ 1
1 + 2 I e0.0625 J, + 2 I e0.25 I + 2 I e0.3625 I + e
* 0.742 98.... •
Although we do not know a function, we can still evaluate its
integral if we have sets of points relating x and y.
Given:
x
4.66
5.72
6.30
6
6.33
5.85
8
5.23
Use the trapezoidal rule to approximate the value of the integral
of the function f, defined by the above points.
n = 5
3
7
I
a. -
0
O
A.
4
3-
9
Z
1
X
-f
< \-i
,x)
<
I i
3 4
t
> e
■> r
J i
3 9
x
8
8-3
j/X*) dx * —— (4.66 + 2 • 5.72 + 2 . 6.30 + 2 . 6.33 + 2 • 5.85 + 5.23)
2 • 5
* 29.145

952
Chapter 32 METHODS OF APPROXIMATION
Boundary Estimation and Convergence
The error R of an approximate integral T may be defined as
R = T-I,
where 7 is the true value of the integral.
If T < 7, R is negative;
if T > 7, R is positive.
In an interval where the curve of a function f is concave
downward and, consequently, its second derivative is negative, the
error is always negative irrespective of the location of the
curve above or below the x-axis:
y
A
R VTA
fix)
I
T
R
f"
>
>
<
<
<
0
0
7
0
0
X
R ES3
\
X
fix)
I
T
R
f"
<
<
<
<
<
0
0
7
0
0
y=fix)
Analogously, when the curve is concave upward the error is
positive; the error always has the same sign as the second
derivative of the function.
y
A
T K53
R ESS
y=fix)
fix)
I
T
R
f"
>
>
>
>
>
0
0
7
0
0
X
y
A
a
—>-x
Ax) < o
/ < 0
T > I
R > 0
f" > 0
T IZZZ
R rsvq
The closer y = f(x) agrees with a straight line, the smaller
f' \x) becomes; the magnitude of the error can be expected to be
proportional to the magnitude off1 *(x). Therefore, the lower
boundary of T must be a factor of the minimum value of f"(x)
and the upper boundary a factor of the maximum value of
f' \x), which leads to the boundary formula of the trapezoidal
rule,
_ (6-a)3 , _ f* . , . TT (6-a)3
LT _ Q < Tn - J fix) dx < UT
a
12 n2
12 n2
where Lt is the lower boundary value and Ut is the upper
boundary value of /*"(*) on the interval [a, 6].

Section 32.4 Numerical Integration 953
The trapezoidal rule for n = 4 has given
l
IV*2 dx * 0.742 98...;
o
determine the number of reliable decimals.
Let f(x) = IV*2 dx; then /" (*) = - 2 x e"*2 ;
/*"(*) = (4 x2 - 2) e"*2 ;
_v2
f "(*) = 4x (3-2x2)-e~xZ.
In [0, 1] the minimum value of f ' (x) is /*' ' (0) = -2; the
maximum value is f ' (1) = 2 e_1; fu,(x) > 0, that is, the curve
ofy =f,l,(x) is located above the x-axis. Consequently, f"{x)
is increasing and /*"(0) = -2 and /*"(1) = 2 e_1 must be the
upper and lower boundary values, respectively.
Use the boundary formula of the trapezoidal rule:
l
a-o£ ^ 074298_ re_*2 ^ < 2e_i. 11^
12 • 42 QJ 12 • 42
1
-— < 0.742 98- IV*2 dx <
96 J 96 e '
o
Multiplying by (- 1) gives
l
"55~* fe-*2dx - 0.742 98 < ^-,
96 e J 96
o
which we rearrange to
l
- ^- + 0.742 98 < IV*2 dx < ^ + 0.742 98,
0
1
0.739 14 < IV*2 dx < 0.753 40 .
o
Since
and
0.742 98 - 0.739 14 = 0.003 84
0.753 40 - 0.742 98 = 0.010 42
and the absolute error must not exceed a half-unit of the digit,
the approximation 0.742 98 is reliable only to the first decimal
place.
Hence,
[e-*2 dx * 0.7 .
o

Chapter 32 METHODS OF APPROXIMATION
The trapezoidal rule may be programmed for computer use,
which makes investigation for convergence simple. When
satisfactory convergence can be demonstrated, there is
usually no need to determine the boundary.
It is helpful to use a program that successively doubles the
number of sub-intervals until the results repeat themselves; a
computer programmed to truncate the result at 8 significant
digits and to cease operation when two successive values had
identical digits gave the following results:
Integral
1
IV*2 dx
o
yj
Number of Intervals
1
2
4
8
16
32
64
128
256
512
1024
2048
4096
8192
Approximate Value
0.683 939 72...
0.731370 25...
0.742 98410...
0.745 865 61..
0.746 584 60..
0.746 764 25..
0.746 80916..
0.746 820 39...
0.746823 20...
0.746823 90...
0.746 824 07...
0.746 82412...
0.746 82413...
0.74682413...
- 9Ay trapezoidaC act is done and, as a finaC demonstration of the
simplicity of complexities in numerical calisthenics, I give you
fMr. Simpson for his parabolic roller coaster.

Section 32.4 Numerical Integration
955
SIMPSON Thomas
(1710-1761)
Simpson's Rule
While the trapezoidal rule uses the principle that a chord is the
easiest approximation of a curve extending between two
specified points, Simpson's rule is based on the idea that the
simplest curve which can be made to agree with three arbitrary
points is a parabola. Thus, instead of sums of areas of
trapezoids, Simpson's method uses sums of areas under parabolas to
determine approximate values of definite integrals. Since a
parabola generally approximates better than a straight line to
the shape of a curve, we can expect better results by Simpson's
rule than by the trapezoidal rule.
The rule is named after the English mathematician Thomas
Simpson.
Deriving the Definite Integral of a Quadratic Polynomial
A function
p(x) =Ax2 + Bx + C,
where A, B, and C are constants and A * 0, represents a
parabola:
y
A
y =p(x) =Ax2 +Bx + C
X
The area contained between the arc PQ, the x-axis, and the
ordinates at a and b is given by
\ (A x2 + B x + C) dx.
a
To fit a parabola to the curve y = fix) between a and 6, we choose
the point
a + b
c =
2 '
Give the parabola a coordinate system of its own whose origin
is the point [c,p (c)]. t is the independent variable of the new
coordinate system,
is ^~ *V ~™~ O •
We must now define limit values for the new coordinate
system.

Chapter 32 METHODS OF APPROXIMATION
If (- h) and h are the lower and upper limits, we have
-h = a-c\ h = b-c .
We have the equation
p(x-c) =p(t) = At2 + Bt + C
and the relations
p(-h) = A{-h)2 + B{-h) + C = Ah2-Bh
p(0) = A02+£0 + C = C = qd
p(h) = Ah2 + Bh + C = q(b)
= q(c) I
which give
q(a) + q(b) = 2Ah2 + 2C
q(c) =C
to be inserted in the equation
h
\(At2 + Bt + C)dt =
A t* Bt2 „ '
-h
h
-h
= ^h[(2Ah2 + 2C) + 4C]
= - h [q (a) + q (b) + 4 q (6)] .
Since
J (A (t + c)2 + B(t + c) + C) dt = \ (A x2 + B x + C) d*
-A
a
and
6-a a+6
h = ~T"' C=~2~
we can write
r 1
J(Az2+J3z + C) dx = ^ (b-a)
q (a) + q (b) + 4 q
ra + b
{ 2
a
which we may use to find an approximate value of any definite
integral.
x

Section 32.4 Numerical Integration
957
j>(*)
d* s=»
a
b
b-
— a
6
-a
6
p (a)+p (6) + 4p
f(a) + f(b) + 4f
I 2 Jj
a+ 6^
2 Jj
In the graph below, [a, 6] is divided into an even number n of
sub-intervals with equal length, . As long as the arcs of
the function corresponding to each of the sub-intervals are
consistently concave downward or concave upward, we may
expect the approximation of the curve to an arc of the parabola,
represented in the graph by the quadratic functions pi to pn/2, to
be better the shorter the intervals are.
Pntifi)
x
Since an integral is equal to the sum of its parts,
6 *2 *4 b
[/fa) dx = |/fa) dx + |/fa) dx +...+
a a X2 xn_2
we obtain
J fix) dx,
\f(x) dx & S1 + S2+ ... + Sn
a
= g (x2-a) l/to) + f(x2) + 4/ (xi)]
+ g (X4-X2) 1/(^2) + /(#4) + 4/(^3)] +...
+ q (b-xn_2) \f(xn-2) + fib) + 4/ (xn-i)] ,
which gives us Simpson's rule,
j>fa)
dx &
b - a
3 n
\f(a) + 4 /fai) + 2 /fa2) + 4 /fas)
a
+ 2f(x4)+... + 2f(xn_2) + 4f(xn-1)+/(&)] ,
where n is a positive even integer, indicating n equal sub-
intervals on [a, 6].

958
Chapter 32 METHODS OF APPROXIMATION
Determine an approximate value of the integral
j
o
e~*2 dx
by Simpson's rule for n = 2 and n = 4.
It is understood that no guarantee can be given regarding
reliable digits.
71 = 2
x
0
JV*2d* * ^-^ (e-o +4e-! +e"4)
3 -2
If 4 1
'l + - + —I =: 0.82994
V
e e4
n = 4
j
o
e"*2 dx * —-9- (e- ° + 4 e"1/4 + 2 e" ! + 4 e" 9/4 + e"4)
3-4
= 0.88181....
x

Section 32.4 Numerical Integration
959
When convergence can be satisfactorily demonstrated for
Simpson's rule, there is usually no need to determine the
boundary values. The table below was obtained with a
computer programmed to truncate the result at 8 significant
digits until two successive values are identical:
Integral Number of Intervals Approximate Value
j
0
e~* d x
2
4
8
16
32
64
128
0.829 944 47...
0.881812 43...
0.882 065 51...
0.882 080 40...
0.882 08133...
0.882 08139...
0.882 08139...
The steady convergence towards 0.882 08139... is a strong
indication that the value is correct to at least the 7th place of
decimals.
Boundary Estimation and Convergence
Generally, Simpson's rule leads to a satisfactory value faster
than the trapezoidal rule. On the other hand, error estimation
is a much more formidable task with Simpson's rule than with
the trapezoidal rule.
With Simpson's rule the magnitude of the error is proportional
to the magnitude of the 4th derivative of the integrand, which
leads to the boundary formula of Simpson's rule,
Ls±^t<Tn_ L)^ < Us <Lj£t
180 re4 J 180 re4
a
where Lg is the lower boundary value and Us is the upper
boundary value of f^ in the interval [a, b], and Tn is the
approximate value of the integral evaluated by Simpson's rule
over n sub-intervals.
We try to avoid the use of the boundary formula of Simpson's
rule, for we must find not only the 4th derivative of f but - in
order to determine upper and lower boundary values of
f^ - also the 5th derivative of/*, which can be a formidable
task.

Chapter 32 METHODS OF APPROXIMATION
Like IV*2 x, the indefinite integral fsin(x2) dx cannot be
expressed by elementary functions. The table below compares
computer results obtained with Simpson's rule and with the
trapezoidal rule.
Integral
10
J sinOe2) dx
0
Number of
Intervals
1
2
4
8
16
32
64
128
256
512
1024
2048
4096
8192
16 384
32 768
65 536
131072
262144
524288
Approxinu
Simpson's Rule
-1.726 287 74...
-1.733 979 05...
5.896 28191...
1.270177 02...
0.515 778 43...
0.543 626 35...
0.581723 28...
0.583 575 75...
0.583 665 27...
0.583 670 55...
0.583 670 88...
0.583 670 90...
0.583 670 90...
ite Value
Trapezoidal Rule
-2.531828 21...
-1.927 672 85...
-1.782 402 50...
3.976 610 81...
1.946 785 47...
0.873 53019...
0.626102 31...
0.592 818 04...
0.585 886 32...
0.584 220 53...
0.583 808 05...
0.583 70517...
0.583 679 47...
0.583 673 04...
0.583 67144...
0.583 67103...
0.583 670 93...
0.583 670 91...
0.583 670 90...
0.583 670 90...
y
A
1--
y = sin (jc2)

961
Chapter
33
PROBABILITY THEORY
33.0
33.1
33.2
33.3
33.4
33.5
33.6
Introduction
History
The Basics
The Probability Density Function
Central Tendency
Dispersion
Normal Distribution
Page
962
963
966
971
974
976
979

962
Chapter 33 PROBABILITY THEORY
33.0 Introduction
Probability theory is a branch of statistics, a science that
employs mathematical methods of collection, organization, and
interpretation of data, with applications in practically all
scientific areas.
When working with probability theory, we analyze random -
Greek, stochazesthai, or stochastic - phenomena and assess the likelihood that an
"to guess at", "to aim at" event will occur.
In this chapter we discuss some fundamental aspects of
probability theory and explore its use of integral calculus.
Wheel of Fortune
Gregor Reisch, Margarita phylosophica (first edition, 1496;
1503 reprint).

963
33.1 History
CARDANO Gerolamo
(1501 -1576)
PASCAL Blaise
(1623 -1662)
FERMAT Pierre de
(1601 -1665)
HUYGENS Christiaan
(1629-1695)
BERNOULLI Jakob
(1654 -1705)
One of the earliest mathematical studies on probability was
Liber de ludo aleae ("On Casting the Die"), written by the
16th-century Italian mathematician and physician Gerolamo
Cardano; it was not published until 1663, 87 years after
Cardano's death. Cardano introduced concepts of
combinatorics into calculations of probability and defined probability
as "the number of favorable outcomes divided by the number of
possible outcomes". It is likely that Cardano would be known
as "the father of the theory of probability" had the publication
not been delayed.
In the 17th century, questions about the probability of events
occurring in games of chance were discussed in
correspondence between the French mathematicians Blaise Pascal
and Pierre de Fermat. Building on their results, the Dutch
physicist-astronomer-mathematician Christiaan Huygens
published, in 1656, De ratiociniis in ludo aleae ("On
Reasoning in Games of Chance").
The Swiss mathematician Jakob Bernoulli was an early
advocate of the use of probability theory in medicine and
meteorology in his work Ars conjectandi ("The Art of Conjecture"),
published posthumously in 1713.
JACOBI BERNOULLI,
ProfefC Bafil. & utnufque Sociec Reg. Sckntiar.
Gall. & Prufi: Social.
Mathematics Celeberrimi,
ARS CONJECTANDI,
OPUS POSTHUMUM.
Accedit
TRACTATUS
DE SERIEBUS INFINITIS,
Et E p i s t o l a Gallicc fcripta
DE LUDO PILjE
RETICULARIS.
BASILED,
Impends THURNISIORUM, Fratmm.
do Idcc XI II.

Chapter 33 PROBABILITY THEORY
LAGRANGE Joseph Louis
(1736-1813)
LAPLACE Pierre Simon de
(1749-1827)
GAUSS Carl Friedrich
(1777 -1855)
POISSON Simeon-Denis
(1781 -1840)
de MOIVRE Abraham
(1667-1754)
CHEBYSHEV Pafnuty Lvovich
(1821-1894)
MARKOV Andrei Andreevich
(1856 -1922)
LYAPUNOV
Alexandr Mikhailovich
(1857 -1918)
EINSTEIN Albert
(1879 -1955)
RUTHERFORD Ernest
(1871 -1937)
CHARLIER
Carl Vilhelm Ludvig
(1864 -1934)
PEARSON Karl
(1857 -1936)
FISHER Ronald Aylmer
(1890-1962)
In the late 18th century, it became increasingly evident that
analogies exist between games of chance and random
phenomena in physical, biological, and social sciences.
Contributions of fundamental importance to probability theory
were made in the latter half of the 18th century and the
beginning of the 19th century by the French mathematicians,
astronomers, and physicists Joseph Louis Lagrange and Pierre
Simon Laplace, the omnipresent German mathematician and
astronomer Carl Friedrich Gauss, and the French
mathematician Simeon-Denis Poisson. The most important publication
on probability theory in this era is Laplace's Theorie analy-
tique des probabilites (1812), which discussed practical
applications of the theory and developed the concepts of normal
distribution, first discovered by Abraham de Moivre. In
the 1837 Recherches sur la probability des jugements...
("Researches on the Probability of Opinions...") Poisson
introduced what we now know as the Poisson distribution, or
Poisson law of large numbers, an approximate method used to
describe probable occurrence of unlikely events in a large
number of unconnected trials.
About 1850 - 1900, the Russian (Petersburg) school of
probability theory, emphasizing stringent mathematical methods,
dominated its development. Prominent figures of this
school were Pafnuty Chebyshev and, originally disciples of
Chebyshev's, Andrei Markov and Alexandr Lyapunov.
In the beginning of the 20th century, the need for applications
of probability theory increased in physics, economics,
insurance, and telephone comunication. Important impulses were
given by Albert Einstein, the New Zealand-born English
physicist Ernest Rutherford, and the Swedish astronomer
C. V. L. Charlier. Applications often precipitated new
probability problems which had to be tackled within the field of
theoretical probability, and thus a fruitful interplay between
the sciences was created.
The English mathematician Karl Pearson is the founder of
modern hypothesis testing - he developed the chi-square
test of statistical significance. Besides making major
contributions in mathematics and probability theory, Pearson also
practiced law, was active in politics, published literary works,
and wrote The Grammar of Science (1892), a classic in the
philosophy of science.
One of the most eminent scientists of the 20th century, the
English geneticist and statistician R. A. Fisher, professor of
genetics at Cambridge from 1943 to 1957, developed methods of
multivariate analysis - analysis of problems involving more
than one variable - and used them in his investigations of the
linkage of genes to various traits. He also introduced the idea
of likelihood in statistical inference, that is, how to draw
conclusions on the basis of the relative probability of different
events.
Fisher's Statistical Methods for Research Workers (1925) was
used extensively as a textbook and a reference book and
remained in print for more than 50 years.

Section 33.1 History
965
Statistical Methods for
Research Workers
tr
R. A. FISHER, M.A.
ftlln t/ Gtmnlit mU Catut CaiUgt, Camin<tjt
Cku/ S/a/utiaan, Rtthwiud Exptrimtnt Station
OLIVER AND BOYD
EDINBURGH: TWEEDDALE COURT
LONOOM: \% PATERNOSTER ROW. E.C
I92S
I
INTRODUCTORY
L The Scope of Statistics
The science of statistics is essentially a branch of
Applied Mathematics and may be regarded as
mathematics applied to observational data. As in
other mathematical studies the same formula is equally
relevant to widely different groups of subject matter.
Consequently the unity of the different applications
has usually been overlooked, the more naturally
because the development of the underlying
mathematical theory has been much neglected. We shall
therefore consider the subject matter of statistics
under three different aspects, and then show in more
mathematical language that the same types of
problems arise in every case. Statistics may be regarded
as (i.) the study of populatioas, (u.) ** th« study
of Tmriition, (in.) as the study of methods of the
reduction of data.
PEARSON Egon
(1895 -1980)
NEYMAN Jerzy
(1894 -1981)
KHINCHIN
Alexandr Yakovlevich
(1894 -1959)
KOLMOGOROV
Andrey Nikolayevich
(1903 -1987)
Theory of Games
von NEUMANN John (Johann)
(1903 -1957) p. 301
MORGENSTERN Oskar
(1902 -1977)
Along with Egon Pearson (son of Karl Pearson) and Fisher,
Jerzy Neyman was one of the principal founders of modern
statistical analysis. Neyman lived in Poland until he was
forty, then in England, and finally settled in the United
States. In 1955, he became professor of statistics at the
University of California at Berkeley, where his department
became a world center for the development of mathematical
statistics. Egon Pearson and Neyman founded what is now
known as the school of statistical inference.
The Russian mathematicians Alexandr Khinchin and A. N.
Kolmogorov are the founders of the Moscow school of
probability theory, one of the most influential in the 20th century.
One may say that the present golden age of probability
theory started in 1933 with Kolmogorov's Grundhegriffe der
Wahrscheinlichkeitsrechnung ("Foundations of Probability
Theory"). Kolmogorov introduced several fundamental
postulates in statistics and probability theory; he showed that
probability theory may be founded on the concepts of set theory
and mathematical measure theory.
The theory of games was founded by John von Neumann,
preeminent 20th-century innovator in many fields of pure and
applied mathematics. He created a mathematical model for
games of chance, such as poker and bridge, that involve free
choices - strategy - for the players. His first paper on this
subject was presented in 1926; von Neumann's theories were
further developed in his major work Theory of Games and
Economic Behavior (1944), co-authored with the economist
Oskar Morgenstern. The theory of games is now a
mathematical discipline of its own, with far-reaching applications to
economics and social sciences.

966
Chapter 33 PROBABILITY THEORY
33.2 The Basics
A stopped clocf^shoivs the correct time twice every 24 hours.
Defining Probability
The classical definition of mathematical probability is the
ratio of the number of favorable outcomes to the total number of
possible outcomes.
The total probability of favorable and unfavorable outcomes
is 1; if there is no probability of favorable outcomes, the
probability is 0. If there are 9 marbles in a box,
2 red, 3 green, 4 yellow,
the probability of drawing a red, green, or yellow marble from
the box is 1, whereas the probability of white is 0.
The first time, the probability of drawing
2
a red marble
a green marble
a yellow marble
is
is
is
9
3
9
4
9
1
3
the probability of drawing a red or a green marble is
9 +3 "9
In two successive draws, one marble at a time and putting the
marble back after the first draw, the probability of drawing two
red marbles is n
9 x9
81
= 0.049
If the first draw is not replaced, the probability of drawing two
red marbles is
9 x 8 = 36 = °-027-- •
Conditional Probability
Draw two cards from a deck of cards; what is the probability
that the second card will be the seven of spades?
There are several answers to this question:
- If we have not looked at the first card drawn but simply laid
it aside, the probability for the second card will be the same
as for the first card, that is, 1/52.
- If we looked at the first card, there are two possibilities:
If the first card was not the seven of spades, the chance that
the second card will be is 1/51.
If the first card was the seven of spades, the probability for
the second card is nil.
This simple example shows that "probability" should always
be viewed in the light of what information there is at hand.

Section 33.2 The Basics 967
%
\
- I'd see you. Whal've you got?
- Jive aces — and you?
- Two revolvers.
- 51(1 right, you win.
The Law of Large Numbers
When the number of observations of an experiment repeated
under identical conditions is sufficiently large, then - by the
law of large numbers - the proportion of some specific outcome
will be close to the underlying probability of that outcome; the
greater the number of observations, the closer the agreement.
Conclusions can never be drawn with absolute certainty from
statistical data; they only provide evidence about the
likelihood that a conclusion is correct. Suppose a die is cast 100
times and a 6 turns up every time; if the die is well balanced,
the probability of this happening is
or rather something has happened which would be very
unlikely if the die is not weighted.
Random Numbers
In a set of numbers, e.g., 0, 1, 2 ... 9, each random number has
an equal chance of being selected, and each selection is
independent of all prior selections. To select numbers at random,
we may write each number of a set on a piece of paper which we
then place in an urn; each time a number has been drawn, we
return it to the urn and mix. The sequence
6,7,8,8,6,4,7,1,6,8,6,8,3,2,1
is equally as likely as the sequence
1,2,3,4,5,6, 7,8,9,0,1,2,3,4, 5.
There are tables of random numbers; computers may be
programmed to generate such numbers.

Chapter 33 PROBABILITY THEORY
Monte Carlo Methods
Processes that are too complicated to allow exact analysis may
often be solved by probabilistic methods that employ the law of
large numbers. Named after the famous gambling casino,
they are known as Monte Carlo methods. They are used
anywhere from estimating the strength of a hand in bridge
to modeling the statistics of a nuclear chain reaction. Key
inventors of the Monte Carlo methods were John von
Neumann and the Polish mathematician Stanislav Ulam.
A Composite Area Problem
To determine the approximate area of a plane figure of
irregular or complex outline, we may
o enclose that area by a figure of known area;
o place a given number of random points within the
enclosing figure;
o find the number of points that hit the sought area;
o then, with p being the probability that a random point will
hit the sought area (or areas),
sought area points inside sought area
^ ~ known area points inside known area
Example:
An oil spill breaks up over the sea in smaller and larger
sections. An aerial photograph is taken of the area and it is
estimated that the sections of oil and open water can be
enclosed by a 4 x 5-kilometer rectangle.
- Find the total surface area of the oil sections.
The photograph may be digitized and fed into a computer,
programmed to give the number of random points that hit the
sought area. The greater the number of total points, the more
reliable the estimate.
The area enclosed is 4 x 5 = 20 km2.
We might obtain:
Number of points:
total in the oil sought area (km2)
10 6 ^ — • 20 * 12 km2
100 61 * 100 ' 2° * 12'2 km2
441
1000 441 * 1000 "20 * 8.8 km2
4654 ^ „oi 9
10 000 4654 * 10000 * 9.3 km2
48 581 ^ „r,, 9
100000 48581 * 100000 ' *

Section 33.2 The Basics
969
i rr ■■•;.■:.:*, :.# v.- ^-;vj/.:'^^:.-v:.' .^-:-.^---:- ^-^../ -..----./ ^^-".yr<"^A
Total: 10 000 dots.
Binomial Theorem:
pp, 140,775
Binomial Distribution
If there are two possible outcomes of an event (e,g,, success or
failure), and the possibilities of the outcomes are independent
and constant, the distribution of probabilities is called a
binomial distribution.
Examples of binomial distribution are probabilities regarding
the number of heads or tails when tossing a coin and of red or
black when cutting and reshuffling a deck of cards,
p. 196 The notations nCk and \k), denoting combinations, give the
number of ways one can select k objects among n,
disregarding order.
The probabilities of the binomial distribution are given by the
terms in the expansion of (p + q)n:
(p + q)n = (o)p" + (i)p"-19 + Op"-2?2*--
+ ( nn-1)pqn-1 +CV -
In a binomial distribution, where the probability of success is p
and that of failure is q = 1 -p, the probability of k successes is
nCk pk qn~ k -
(n - k) ! k !
»-j/v jit Th fv
What is the probability of a run of 4 boys in a family with 4
children?
4C40,54(l-0.5)4-4 =
4 !
(4 - 4) ! 4 !
0,54(l-0.5)4"4 = 0.062 5 ,
What is the probability of a run of 4 boys and 4 girls in a
family with eight children?
Comparing with the previous problem, we find that the
probability is
0,0625 (0,0625) = 0,003 906 25 ,

Chapter 33 PROBABILITY THEORY
What is the probability of a run of 8 girls in a family with 8
children?
8C8 0.58 (1-0.5)8 - 8 =
8 !
0.58 (1-0.5)8 - 8
(8 - 8) ! 8 !
= 0.00390625;
thus, the same probability as in the previous example.
In a population, 32% of people have blood group A and 68% have
one of the other blood groups, B, AB, and 0. Five people are
available for an emergency blood donation.
- What are the probabilities that 0, 1, 2, 3, 4, 5 of the possible
donors are blood group A?
Number of donors
with blood group A
0
Probability
5 !
(5 - 0) ! 0 !
5 !
(5 - 1) ! 1 !
5 !
(5 - 2) ! 2 !
5 !
(5 - 3) ! 3 !
5 !
(5 - 4) ! 4 !
5 !
(5 - 5) ! 5 !
0.32° (0.685 - °) * 0.1454
0.32^0.685- !) * 0.3421
0.322(0.685" 2) * 0.3220
0.323 (0.685 " 3) * 0.1515
0.324 (0.685 " 4) * 0.0356
0.325 (0.685 " 5) * 0.0034
The Null Hypothesis
A hypothesis that is being tested for rejection is known as a
null hypothesis - an assumption that observations are due to
chance alone.
The rejection of a null hypothesis when in fact true is known
as a type I error; if accepted despite being false, a type II error
is committed.
Significance Level
The significance level of a test is the probability of committing
a type I error. The lower the significance level, the lower the
probability of a type I error and the stronger the evidence
against the hypothesis. If we can reject the null hypothesis at
the decided-upon significance level, we say that our results are
significant, and our hypothesis is rejected.
(Beyond (Dou6t
- Has it been proved that women live longer than men?
- yes, at kastfor widows.

971
33.3 The Probability Density Function
Random Variable
A random variable is a numeric quantity that can be
measured in a random experiment. More precisely, it is a
function of the possible outcomes of the experiment.
The probability distribution of a continuous random variable x
is a function of the numerical values of x; such a function is
referred to as a probability density function, and it is so
constructed that the area between its curve and the x-axis when
calculated over the entire range of x for which f(x) is defined
equals 1.
If f(x) is a probability density function of a continuous
random variable x that can assume values between x = a
andx = b, then
J f(x) dx = 1
a
The graph of a probability density function may take several
forms; e.g.,
g(x)
x
a
x
hi
i
a
x)
i
i
i
b
X
X
fix) = 4 x3 for 0 < x < 1
is a probability density function, because fix) is positive for all
real values of x, and
l l
J fix) dx = J 4jc3 dx = 4
0 0
^
V4
0
= 1.
With a probability density function
fix) = 4xil-x2) for 0<x<l,
determine the probability that the random variable falls in
[0.2, 0.4] .
0.4
J 4 x (1 -x2) dx
0.2
0.4
0.2
X'
= 4
/0.16 0.0256^ . f0.04 0.0016
LV
V
= 0.216 .

972
Chapter 33 PROBABILITY THEORY
The sought probability is 0.216, corresponding to the shaded
area of the graph:
= 4*(1 -x2)
1.5 -
x
0.0 0.2 0.4 0.6 0.8 1
The time for a certain type of surgical suture to be absorbed by
the tissue of the human body varies according to the probability
density function
fix) = 0.001 e-°001/l; h>0,
where h denotes hours.
What is the probability that a suture will remain in the tissue
a: for no more than 600 hours;
b: beyond 1000 hours.
a:
600
J 0.001 e-°001h d/i =
o
600
0
(_e-o.ooi*) = i_e-o.6 ^ o.45.
0.0010
0.0008
0.0002
y=f(h) = 0.001 e-0001h
0.0000
0 500 1000 1500 2000 2500 3000
*»h
The sought probability for case a is 0.45 .

Section 33.3 The Probability Density Function
973
b:
1000
1- J 0.001 e-omih d/i
o
= 1- [-e
- 0.001 h
]
1000
0
= l-(l-e-01) * 0.37
y
0.0006 -
0.0004 -
0.0002
0.0000
^ = /(/0 = 0.001 e-°-001/l
►*»*»*X*»*»*X,»*»***»*»*X,»,»*»*»*»*»,**»"»*»''»*»*»*»*»*»*»*i
mmp^szumpx
0 500 1000 1500 2000 2500 3000
+-h
The sought probability for case b is 0.37.
ERA OF REDUCED EXPECTATIONS
WELL, IP EV6RYTU1MG-
i C-06S O.K.,THAT OLD
J Jl
LEFT FOOT SHOULD BE
V COMJKlCr INTO VIEW SOON
5N
'••'-■••*■' v -
Ji
is
N\AMKotF
Drawing by Mankofff; ©1989
The New Yorker Magazine, Inc.

Chapter 33 PROBABILITY THEORY
33.4 Central Tendency
A measure of central tendency is a summary of a whole
distribution of events or measurements. The three most useful
measurements are the mode, the median, and the mean.
Mode
The value of events or experimental data that occur most
frequently is the mode of the data.
The mode of the values 1, 3, 3, 3, 5, 5, 7, 9, 9 is 3; the modes of
2, 2, 2, 3, 3, 9, 4, 9, 9,1, 2, 9, 5 are 2 and 9.
Playing with two dice, the sum of the throws may vary from 2 to
12. What would be the expected mode after a large number of
throws of the dice? What is the probability, each time, of
throwing a seven, a two, or a twelve?
1 + 1
2 + 1
3 + 1
4 + 1
5 + 1
6 + 1; 2; 3; 4; 5; 6 = 7; 8; 9; 10; 11; 12
Sums of the dice: 1 two; 2 threes; 3 fours; 4 fives; 5 sixes;
6 sevens; 5 eights; 4 nines; 3 tens; 2 elevens; 1 twelve.
Sum total: 36
The mode is seven.
The probability of throwing a seven is 6/36 & 0.167; that of
throwing a two is 1/36 & 0.03, as is that of throwing a twelve.
; 2,
; 2
; 2
; 2
; 2,
; 2
3,
; 3
, 3
, 3
3
; 3
4,
, 4
. 4,
, 4
, 4
, 4
5,
, 5
5,
, 5
, 5,
; 5
6 =
, 6 =
6 =
, 6 =
6 =
, 6 =
= %
= 3
= 4,
= 5
= 6,
= 7,
3;
, 4,
5;
, 6
7
8
4;
5;
6;
, 7,
; 8
; 9
5;
6;
7;
8;
; 9;
; 10;
6;
7;
8;
9;
10:
7
8
9
10
11
Median
The median is the middle value of a sequence of observations
arranged in order of magnitude; the number of values greater
than the median equals the number of values smaller than the
median. If the number of observations is even, the median is
the arithmetic mean of the two middle observations.
For instance, the median of 7, 9, 72, 82, 100, 101, 119 is 82; the
median of 3, 9, 11, 14, 16, 19, 23, 27 is 15.
For a continuous random variable x with probability density
function f over [a, b], the median of x is the number m for which
m b
J f(x) dx = J f(x) dx = — .
a
m

Section 33.4 Central Tendency
975
Mean
When simply referring to the mean, or average, we usually
have in mind the arithmetic mean:
The arithmetic mean jll of x\, x<i... xn is
n
n
— j^^ Xi — v^l + X% + • • • + Xftj
i = l
Weighted Means
If in 120 observations the numbers
0, 1, 2, 3, 4, and 5
occur, respectively,
0, 8, 16, 24, 32, and 40 times,
then
0-0 + 8-1+ 16 -2 + 24 -3 + 32» 4 + 5-40 11
M = 120 = T * 367 *
The probabilities of the finite values of observations can be
expressed using the function
r 0/120 when xt = 0
8/120 Xi = 1
16/120 xt = 2
24/120 xt = 3
32/120 xt = 4
v 40/120 xt = 5
/x**) = <
and the arithmetic mean as
= 0 (0/120) +1 (8/120) + 2 (16/120) +3 (24/120) +4 (32/120) +5 (40/120)
= — * 3.67 . •
Weighted means give mark to the importance of different
data.
If entrance to an educational program weighs English as 4,
Mathematics as 3.5, Physics and Chemistry as 3, Biology as 2,
and all other subjects as 0, then a student with a 4 in English, 3
in Mathematics, 2 in Physics, and 3 in Chemistry and Biology
has the weighted arithmetic mean /iw :
_ 4-4 + 3.5-3 + 3-2 + 3-3+ 2-3 _
Mw " 4 + 3.5 + 3 + 3 + 2 " 306
Mean of a Continuous Random Variable
For a continuous random variable x with probability density
function f over [a, 6], the arithmetic mean jll of x is
jii = x f(x) dx.
a

976 Chapter 33 PROBABILITY THEORY
33.5 Dispersion
Range
The simplest measure of dispersion, or spread, of data is
the range, that is, the difference, or distance, between the
highest and lowest observed values. When dealing with the
extreme values, the range is a rather rough value of the
dispersion.
Variance and Standard Deviation
Taking into account the deviation of every observed value
from the mean, the variance and the standard deviation are
sensitive indicators of the degree of variability of statistical
observations. The standard deviation is simply the square
root of the variance.
If /i stands for the mean, we have for a data set:
o the variance <72 (lowercase sigma),
2 K* - M)2 .
° " n - 1 '
o the standard deviation o\
\ 71-1
The reason for specifying (n - 1) and not n in the denominator
is due to the fact that there are only (n - 1) independent pieces
of information besides the mean. For great values of n, it is
usually of no significance whether n or (n - 1) is used.
Determine the standard deviation of the observations
3,4,5, 6, 7,9,10,12 .
11 = -(3 + 4 + 5 + 6 + 7 + 9 + 10+12) = 7.
o
Calculate the deviation of each of the 8 observations from the
mean; square the results:
(3 - 7)2 = 16
(4-7)2 = 9
(5-7)2 = 4
(6-7)2 = 1
(7-7)2 = 0
(9-7)2 = 4
(10-7)2 = 9
(12 - 7)2 = 25

Section 33.5 Dispersion 977
CHEBYSHEV Pafnuty Lvovich
(1821-1894)
The variance is
a2 = -—=-(16 + 9 + 4+1 + 0 + 4 + 9 + 25) = —
o — 1 /
and the standard deviation is
68
7=" = 3.116 7 ....
Chebyshev's Theorem
If a distribution of measurements has a small standard
deviation, most measurements are likely to be closely grouped
around the arithmetic mean; on the other hand, a large value
of the standard deviation indicates a greater variability, and
therefore the measurements are likely to be more spread out
from the mean.
This idea is formally expressed in Chebyshev's theorem, or
Chebyshev's inequality, given here without proof:
If a probability distribution has the arithmetic mean /i and the
standard deviation a, then, for k > 0, at least 1--^ of the
measurements of any set of observations will have a value
within k a of the mean of the measurements.
p. 967 The law of large numbers is an immediate consequence of
Chebyshev's theorem.
• Consider a random sample of 400 components, manufactured
to a nominal diameter of 18.62 mm, with a mean of 18.64 mm
and a standard deviation of 0.03 mm. To determine the
interval of 300 measurements, we have
400 ( 1-^-1 = 300
k = ±2
and find
18.6412(0.03) = 18.64 ± 0.06 mm.
From the result, we may infer that at least 300 of the 400
machined components have a diameter 18.64 ± 0.06 mm, and
that no more than 100 components have a diameter below 18.58
mm or above 18.70 mm.

978 Chapter 33 PROBABILITY THEORY
z Score
To compare, or rank, observations from two independent sets
of observations or populations, we may convert the individual
observations into standard units, referred to as z scores or
z values.
In a population with the arithmetic mean /i and the standard
deviation a, the z score is given by
X - JlI
a
For jobs as city guides for foreigners, a group of linguistically
inclined candidates was given tests in French, Spanish,
Italian, and German. One of them, Madelaine, who felt that
French was her strong point, was particularly eager to make
use of her French, but would also accept a job as a guide for
Italian tourists. She knew that her German was rather rusty,
but she still had to take the German test.
Results:
Language
French
Spanish
Italian
German
Madelaine's
Mean Standard deviation
550 40
380 40
480 60
300 25
z scores were:
French:
Spanish:
Italian:
530-550
Z " 40
400 - 380
Z " 40
500-480
Z " 60
Madelaine's score
530
400
500
350
1
" ~2
1
2
1
3
350-300
German: z = — = 2
We see that although Madelaine got her highest score in
French, this score was j of the standard deviation below the
mean of the competitors' French scores. As she expected, she
got her lowest score in German, but this score was nevertheless
2 standard deviations above the mean of the German scores.
She was accordingly offered a job as a guide for German
tourists but did not take it.
■y

979
33.6 Normal Distribution
Historical Notes
deMOIVRE Abraham
(1667 -1754)
LAPLACE Pierre Simon de
(1749-1827)
POISSON Simeon-Denis
(1781 -1840)
GAUSS Carl Friedrich
(1777 -1855)
QUETELET Adolphe
(1796-1874)
GALTON Sir Francis
(1822-1911)
In the early 18th century it became apparent to scientists who
studied the distribution of errors in repeated measurements
that observations of a great number of different measurements
often tend to show a similar form of distribution, now called
normal distribution or Gaussian distribution. In 1733, its
mathematical equation was formulated by the mathematician
and statistician Abraham de Moivre, born in France, educated
in Belgium, and finally resident in England. The
mathematical properties of the normal distribution were studied and
explained by Pierre Simon Laplace, Simeon-Denis Poisson,
and Carl Friedrich Gauss.
The Belgian astronomer, mathematician, and statistician
Adolphe Quetelet had studied astronomy and probability with
Laplace in Paris and was the first to apply normal
distribution to the study of sociology. Quetelet presented his concept of
the "average human being" (Vhomme moyen) around whom
measurements of human traits were grouped in normal
probability distributions. His observations of the numerical
consistency of what had been supposed to be voluntary acts of
crime provoked extensive discussions about free will versus
social determinism; such studies are still an important
subject for research on social behavior and criminology.
Noncontroversial, however, was Quetelet's development of
statistical methods for collecting and analyzing large
numbers of simultaneous astronomical, meteorological, and
geodetic observations made at points in Europe chosen at
random.
Francis Galton, English explorer and anthropologist, and an
early authority on fingerprinting, developed Quetelet's
observation of the distribution of certain measurable human
characteristics and became a pioneer in using the normal
distribution in work on heredity.
Normal Distribution Functions
The probability density function of a continuous random
variable with a normal, or Gaussian, distribution is
fix) =
ox2n
; -oo <x < oo 9
where a and fi denote the standard deviation and its arithmetic
mean, respectively.

980
Chapter 33 PROBABILITY THEORY
fix) is of bell-shaped form:
y
A
y = fix) =
<j\[2k
1 fx-jj\
e 2
a
J
x
H (mean)
For fix) to be a probability density function, its integral over
the interval — ©o < x < °o must equal 1.
r
a V 2 7i
l
2
x - n
dx = 1
oo
Why? Letz= -—-; then
a
Therefore, if
we have
dz__l
dx ~ a
and dx = odz
cW 2 7i
*2
^ / \2
_ If X-/J
_ 2
cbc = 1
X\
V 2tc J
-z2/2
dz = 1
or
J e - *2/2 dz = ^2rc
p. 887
To confirm this result, use the method of transforming double
integrals from orthogonal to polar coordinates.
Let
/= J e-*2/2dz.
To obtain a double integral, write
I2 =
f
v
\
J e-^dz
-z2/2
J

Section 33.6 Normal Distribution
981
For z = x and z = y, we have
/2= J e-*2/2d* J e~y2/2dy
oo oo
= f f e-(x2+3,2)/2 ^ dy
— oo — oo
To evaluate this double integral, we transform it to polar
coordinates. To achieve a better understanding of this
procedure, we can think of the above expression as a volume
bounded by the function fix, y) = e~^x +^)/2 an(j ^he infinite
x, y-plane.
We note that the range of x and y from -<*> to °° in rectangular
coordinates is equivalent to an infinite radius sweeping
one complete revolution (2tc rad) about the origin in polar
coordinates.
Since r = V*2 + y2, we have
oo oo
72= J J e-(x2+y2)/2 fa dy
— oo — oo
2k
oo
= \ \ e~r2/2 r dO dr = 2 n \ e~r2/2 r dr
0 0
0
With t = - r2/2,
d£ d£
-7— = -r, or dr = - — .
dr r
r
Iz = 271
e* r
0
_ oo oo
(- -\dt = -2n \ etdt = 2n j e~* dt
^ ' o o
= 2%.

982
Chapter 33 PROBABILITY THEORY
Thus,
I = j e~z2/2 dz = ^2^
and
V 271 J
-z2/2
d2 = 1
Using the substitution z =
we find
X - jl
a
to reinstate the variable x,
oo
V 271
oo
/ A2
f£z£l • \
cbc = 1
and
oo
a V 2 7i
X- \i
V <* y
cbc = 1
— oo

Section 33.6 Normal Distribution
983
Use of Numerical Integration and
Standard Normal Distribution
Exact numerical values may be determined of
1
f(x) =
l
2
' x - \i *
V <* J
oy2n
for the intervals l-^ooj and [0, <»[, but rigorous mathematical
integration for intervals with a finite number of intervals is
not possible.
One way of handling such cases is numerical integration;
another way is to use the substitution z =
f(x) into a standard normal distribution
1
X - Jil
a
and transform
0(2) =
V2Trc
,-z2/2
whose graph is symmetrical about the arithmetic mean,
located atz = 0.
<p(z)
A
1_
X
-5 -4 -3 -2 -1 6
2 3
Tables, such as the one on p. 985, are available for
determining area under the standard normal distribution curve.
The use of such tables dispenses with the need for laborious
numerical integration.
Given a normal distribution with arithmetic mean 9.4 and
standard deviation 3, what is the probability of a random
variable assuming a value between 7 and 12.4?
We have
z1 =
X\ - Jil
a
7-9.4
= -0.8
*2 =
X2-JH
a
12.4-9.4
= 1
Thus, if the probability is p,
p (7 < x < 12.4) = p (- 0.8 < z < 1)

Chapter 33 PROBABILITY THEORY
Alternative I
Using the table for standard normal distribution, we find
0(1) = 0.8413
and
Thus
0(-0.8) = 1-0.7881 = 0.2119.
p(-0.8<z<l) = 0.8413-0.2119 = 0.629
y
Standard normal distribution
y = 0 (z) = —L e-*2/2
V27C
1(^-)
3J2k
Normal distribution
► x and z
7 9.4 12.4
Transformation to Original distribution
standard normal
distribution
Alternative II
p (- 0.8 < z < 1) = —L=r [ e~z2/2 dz
a/0 it J
-{2k
-0.8
p. 949
Using a computer programmed for numerical integration, by
the trapezoidal rule, we find
Integral
V27C
J e-*2/2 dz
-0.8
Number of Intervals
1
2
4
8
16
32
64
128
256
Approximate Value
0.478 496 05
0.596 505 32
0.621435 55
0.627 487 14
0.628 989 49
0.629 364 43
0.629 458 12
0.629 481 54
0.629 487 40

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Chapter 33 PROBABILITY THEORY
Alternative ID
To integrate with respect to x between x\ = 7 and x<i = 12.4, we
may also use the normal density function
f(x) =
1
2
oy2n
' x - \i
Using the trapezoidal rule, we find
(*-¥)
dx
Number of Intervals
1
2
4
8
16
32
64
128
256
Approximate Value
0.478 496 05
0.596 505 32
0.621 435 55
0.627 487 14
0.628 989 49
0.629 364 43
0.629 458 12
0.629 481 54
0.629 487 40
The sought probability is 0.629 .
A type of battery has an average useful service life of 3300
hours, with a standard deviation of 500 hours.
What is the probability that a given battery will last less than
2500 hours?
2500 - 3300 _
Z " 500 --1-6.
The sought probability p is
-1.6
p(x < 2500) = p(z < - 1.6) = —== \ e-*2/2 dz,
i2n
— oo
which we solve by using the table on p. 985,
0(- 1.6) = 1 - 0.9452 = 0.0548 .
The sought probability p
area in the graph.
= 0.0548 corresponds to the shaded
y = 0(z) = -J=re-z*to
Area = 0.0548
-1.8

Section 33.6 Normal Distribution
987
A batch of 4000 screws has a nominal length of 30.00 mm
with a normal distribution of the lengths. A sample of 300
screws was found to have a mean length of 29.99 mm with a
standard deviation of 0.02 mm.
Find: a: How many screws are likely to be 30.00 mm or
more in length?
b: Between what lengths are 99.994% of the screws
likely to be?
a:
The probability of a length of at least 30.00 mm is defined by
30.00 - 29.99
z =
0.02
= 0.50
The sought probability equals the definite integral
oo
k-z2/2
V2~7C
dz,
0.50
which may be solved with numerical integration or using the
table on p. 985. Using the latter method and denoting the
sought probability p,
p (x > 30.00) = p(z> 0.50)
= l-p(z<0.50)
= 1 - 0.6915 = 0.3085 .
This is equivalent to
p (x > 30.00) = p(z> 0.50)
0.50
= 1--^= f e-*2/2 dz,
^2n
— oo
that is, the total area under the curve less the shaded area in the
graph below.
y
A
Area = 0.6915
y = O(z) = -=L e-*2/2
V2ti
-► z
Hence,
0.3085-4000 = 1234,
suggesting that about 1200 screws are likely to have a length
> 30.00 mm.

Chapter 33 PROBABILITY THEORY
b:
The table on p. 985 shows that the area under the standard
normal distribution curve, between z = -4 and z = 4, is
0.99997 -(1- 0.99997) = 0.99994,
equivalent to
4
—= [ e-*2/2 dz * 0.99994 .
Therefore, if the sought boundaries are xi and x<i, then
xi - 29.99
" 0.02
xi = 29.91 mm
and
jc2 - 29.99
0.02
X2 = 30.07 mm.
The result may be verified by
30.07
r / \ 2
0.02 V 2 71
l^x-29.99^
2 V 0.02 /
J
29.91
djc ^ 0.99994.
Hence, 99.994% of the screws will have a length between
29.91 mm and 30.07 mm.
* * *
Taking 0\[p Chance
^en yellow and ten blue socles are in a drawer in a darf^ bedroom.
How many socles should one picf^ to be certain to get:
A. One pair of matching socles?
*B. One pair of a specific color?
Answer: sspos 8AJ8MJ, a s^oos aa-iqj, V
0\[pt "Even Half a Chance!
Wetty is at the fair with 32 dollars in cash. She bets on the toss of
a coin, si^ times. *Betty wagers half her cash every time; her
opponent matches her bets. *Betty wins half the tosses. How much
cash did she have in the end?
JSSUIUUIM
Answer: puB sassoj aq^ jo japjo aq^ jo aArpadsaxn '0S'8I$

989
Chapter
34
DIFFERENTIAL EQUATIONS
Page
34.1 Fundamental Concepts 991
34.2 First-Order Ordinary Differential Equations 994
34.3 Formulating Differential Equations 1000
34.4 Second-Order Ordinary Differential Equations 1015

990
s&Msa^sOfir&Jt

991
34.1 Fundamental Concepts
A differential equation contains one or more terms involving
derivatives of one variable (the dependent variable, y) with
respect to another variable (the independent variable, x), such
as
dy
■• — & x .
ax
Unlike algebraic equations, the solutions of differential
equations are functions and not just numbers. The solution of
the above equation is the integral
y = J2xdx = x2 + C,
where C is an arbitrary constant.
In physics, chemistry, biology, and other areas of natural
science, as well as areas outside natural science such as
engineering and economics, we often encounter the task of
solving the relationships between rates of change of continuously
varying quantities, which is exactly what the terms of a
differential equation represent; thus, differential equations are
essential to all scientific investigation.
Ordinary and Partial Differential Equations
A differential equation that involves a function of a single
variable and some of its derivatives is an ordinary
differential equation. Differentiating
f(x) = y = x3 + 5x2 + 3x + 2
gives the ordinary differential equation
3^- = 3x2 + 10x + 3.
ax
If the differential equation involves functions of two or more
variables and some of their partial derivatives, the equation is
a partial differential equation. If we differentiate
f{x, y) = z = x^ + 3x2y + 3x-2xy5
with respect to x and to y, we get the two partial differential
equations
Y~ = 3x2 + 6xy + 3-2y5 and t— = 3x2- 10xy4.
Order and Degree
The order of a differential equation is the order of the highest
derivative that appears in the equation; consequently,
-p = 3x2 + 10x + 3; rr- = 3x2 + 6xy + 3-2y5; rr- = 3x2-10xy4
are all first-order differential equations.

992
Chapter 34 DIFFERENTIAL EQUATIONS
Degree
The equation
d2y
[3x2 + 10x + 3] = 6x + 10
dx2 dx
is a second-order differential equation; the equations
ym = 6 and y^ = 0
are third- and fourth-order differential equations,
respectively.
The equations
3%
= 6x + 6y ;
3%
= 6x- 10 y4;
3¾
= -40 xy3
3jc2 ' dx dy 3y2
are all second-order differential equations.
The degree of a differential equation whose terms are
polynomials in the derivatives is defined as the highest power of
the highest-order derivative; consequently, the differential
equations
dy
dx
+ 5xy = xz;
d2y dy d2y fdy\3
x —— + -j— = 3; x —r- + -t— = 15
dx2 dx J&2 ^ dLx:^
are first-degree differential equations, and
■ I + 5xy = 2xL\ x
X
dx)
V
dx2
dy
+ dx~7;
x
fd2y
V
dx2
fdyy
K dx ) ~
are second-degree differential equations.
General and Particular Solutions
A solution
Family of Curves
y = J2xdx = x2 + C
dy
is the general solution of the differential equation -r~ = 2 x and
represents an infinite number of solutions, where the
arbitrary constant C may be any real or complex number.
Plotting the solution for different values of C, we obtain a
family of curves, as shown in the graph below for some integer
values of C.
CO »H
II II II
<n o I
II II II
x

Section 34.1 Fundamental Concepts 993
The correspondence of a particular value of the independent
variable x with a certain numerical value of the dependent
variable y will give the constant C a specific value, and this
Particular Solution solution is referred to as a particular solution. If x = 1
corresponds to y = 3, we insert these values in the general solution
y = x2 + C and obtain C = 2.
Thus,
y = x3 + 5x2 + 3x + 2
is a particular solution of all the differential equations
y = 3x2+10x + 3; y" = 6x+10; y'" = 6,
whose general solutions are, respectively,
y = J (3 x2 + 10 x + 3) dx = x3 + 5 x2 + 3 x + Ci;
y = J J (6 x + 10) dx2
= J (3 x2 + 10 x + Ci) dx = x3 + 5 x2 + Ci x + C2 ;
y = J J J 6 dx3 = J J(6x+ Ci) dx2
= J (3 x2 + Ci x + C2) dx = x3 + o"Ci x2 + C2 x + C3 .
Verifying Solutions
Ordinarily, when we refer to the solution of a differential
equation, we mean its general solution.
The solution of a differential equation contains no
derivatives; together with its derivatives, the solution must satisfy
the original differential equation.
To verify a solution of a differential equation, we simply
substitute the solution and its derivatives into the given
differential equation; a correct solution will result in an
identity.
Verify that y = (Cix2 + C2X), where C\ and C<i are arbitrary
constants, is a solution of
x2y"-2xy' + 2y = 0.
y = C\ x2 + C2 x
y ' = 2Cix + C2
y" = 2d
I = 0-x2 + 0-x
2y
-2xy •
x2y"
=
=
=
2 d • x2 + 2 C2 • x
- 4 d • x2 - 2 C2 • x
2 Ci • x2

994
Chapter 34 DIFFERENTIAL EQUATIONS
34.2 First-Order Ordinary Differential Equations
To solve a differential equation, some method must be found of
rearranging or transforming the equation so that we can
integrate its terms.
Directly Integrable Equations
A differential equation of the type
dy
dx
= f(x)
may be solved by integrating both sides directly.
Solve the equations
dy
dx
= 2x + 5\
dy
dx
= 6e3x + 2/x
Direct integration gives
y = J (2 x + 5) dx
= x2 + 5 x + Ci
y =
f
6 e3*
2\
\
dx
x J
= 2e3x +2]nx + C2
where C\ and C2 are arbitrary constants.
Separation of Variables
A differential equation
rty) £-**>,
where f is a function of y only and g is a function of x only,
may be solved by separation of the variables.
dy
We recall that a derivative -r~ can always be written in
differential form; thus,
f(y)dy = g(x) dx.
Iffandg are continuous functions, then
J f(y)dy = jg(x) dx .
• Solve the equation -r~ = x y2 .
Writing in differential form, dy = x y2 dx, and separating the
variables,
r 1 1 xz
— dy = \x dx; --=—+C.
v2 y z

Section 34.2 First-Order Ordinary Differential Equations
995
dy
• Find the general solution of -r— = y sin x, and the particular
solution if y = 1 for x = 0 .
," 1
— dy = J sin x dx
j y
In y = - cos x + C; In 1 = - cos 0 + C; C = 1,
In y = - cos jc + 1 .
dy
• Solve the equation -r— = 4 x (y - 2).
f 1
—j"2 dy = J4x dx; ln(y-2) = 2x2 + Ci
y-2 = eci-e2*2,
and with eci = C, y = 2 + Ce2*2.
Substitution of Variables
First-order differential equations are not always separable.
Homogeneous differential equations, however, may be
transformed into separable equations by the substitution of a
variable.
Homogeneous Differential Equations
The degree of a term is the sum of the exponents of the
variables in the term. An expression is said to be homogeneous
if all terms have the same degree;
x2 y + 3 x y2 - 4 y3
is a homogeneous equation in x and y, all terms being of the
third degree.
Similarly, f(x, y) = V*2 + x y is a homogeneous function of the
first degree, but f(x, y) = V*2 y + x y is not.
2x dx + S^x2+y2 dy = 0
is a homogeneous differential equation, since 2x and Syx^+y2
are both functions of the first degree.
The term homogeneous is also used to indicate that the right-
hand member of a linear differential equation is 0.
In the equation ^ jc + 2 y
dx ~ 3 x '
the variables are not separable in the manner described above,
but they can be made so by introducing the substitution
y = vx,
where v is a function of x.

996 Chapter 34 DIFFERENTIAL EQUATIONS
Differentiating y = v x with respect to x, we obtain
dy dv
dx ax
We can now separate the variables; thus,
dv x + 2 v x dv 1 + 2 i> 1 -1; _
1; + x -5— = — ; x -3— = —-— - v ; x dv = —-— dx,
dx 3 x dx 3 3
and
r
r
1-v
dv =
-— dx
3 x
1 3p-
-ln(l-i>) = — lnx + Ci ; lnCl-i;)-1 = InV* + lnC2,
= Co a/x ; y = x-Cx^3 .
1-v * J
Solve the homogeneous equation
2xy o7 = x2 + 2^2' yd) = o.
Rewrite as
dy x2 + 2 y2
dx 2 x y
Substitute y = vx;
dv x2 + 2 (v x)2 dv 1 _ dx
i> +x -3— = — ; x-7— = -— ; 2y du = —
ax 2 x v x dx 2 i> x
J 2 1; di; = — cbc; u2 = In x + C\ .
./
x
y
Inserting v = — gives the general solution
y2
—— = In x + C; y2 = x2]nx+ Cx2
xz
y(l) = 0 gives C = 0.
Hence, the particular solution is
y2 = x2 In x .
Integrating Factors
A linear differential equation of order n is one that can be
written
d^y dn ~ *y dy
a™ (x) + ctn _ 1 (x) — +...+ a\ (x)-7— + an (x) y = f(x),
dxn dx"-1 dx uv /J /v
where ao (x), a\ (x), ..., aw_i (x), aw (x), and /*(x) are given
functions of x, or constants.

Section 34.2 First-Order Ordinary Differential Equations
997
The linear differential equation of the first order,
^ + P(x)y = Q(x),
arises in many applications of differential equations. Its
solution is simple and straightforward but differs from the
methods described above.
To solve this type of first-order linear differential equation,
Eulers Multiplier we use an integrating factor or Eider's multiplier.
With the integrating factor e^^ , we have
*L (eJ>M d*) + P(x)(eSPM d*)y = Q(x) (eJ/»W dx) f
where the left-hand member is the derivative of y e ^ p ^ ***;
therefore,
and
ye!P(x)dx = JQ(x)elP(x) dx fa
Thus, the general solution of the first-order linear differential
equation d
^- + P(x)y = Q(x)
becomes
• Solve the linear differential equation
t+2y = s-
As an integrating factor, use e^ 2 ^ = e2*,
y = — J3e2*d* = e"2* f| e2* + CJ = |+ Ce~2x. •
A given differential equation must sometimes be rearranged
to the standard form.
• Solve the equation x -:— -y = x3 .
Rearranging the equation, we have
Using
t~ - ~~ y = x
ax x
e! (-x- !) dx _ e~Sx- 1 dx _ e-\nx _ x-l
as an integrating factor, we obtain
y = ——- J x2 x'1 dx = x "5"+ C \ =— + Cx
V

998
Chapter 34 DIFFERENTIAL EQUATIONS
Find the particular solution to x2y' + 2 x y = cos x
71
if y I 2 1 = ° •
Rewrite the equation in standard form,
2 cos x
y + -y =——»
x XL
and introduce the integrating factor
AP{x) (k _ AWx) d* - ~21nx
= e"
= e
= x2
COS X
and insert, with Q (x) = ——, in the general solution:
X'
y =
COS X
X
• x2 dx = —- J cos x dx = —- (sin x + C)
71
y | 2 I = 0 gives C = - 1.
y =
(sin x - 1)
X'
BERNOULLI Jakob
(1654-1705)
BERNOULLI Johann
(1667-1748)
The Bernoulli Equation
dy
dx
+ P(x)y = Q(x);y*
- named after the brothers Jakob and Johann Bernoulli,
second generation of the celebrated but rivalry-filled family
(fathers against sons, brothers against brothers) of Swiss
mathematicians - differs from the equation-^—+ P(x)y = Q(x)
only by the factor yn of the right member.
Convert the Bernoulli equation to linear form by dividing each
term by yn :
y-n-^-+P(x)y1-n = Q(x) .
Introducing the new function z (x) = y *■ ~ n , we have
dz
dy
— =(1_n)y-n^f ory
-n
dy
dx
dz
1 - n dx '
and thus
dz
1 - n dx
+ P(x)z = Q(x) .
Multiplying each term by (1 - n), the original equation
becomes
-^+ (l-n)P(x)z = (l-n)Q(x),
giving the general solution of a Bernoulli equation,
,1- n -
J(l-n)P(*) d
- [(l-n)Q(x)e^1-n)p(x) dx dx.
X J

Section 34.2 First-Order Ordinary Differential Equations
999
Solve the Bernoulli equation
y + -y = xy
We introduce
P = - ; Q = x; n = 2; (1-n) = - 1
and have
J(l-n)
P dx = -
— cbc = -lnx ;
x
,-ln jc
= X
(- 1) • Q • — cbc = — JdLx: = -x + C
y_1 = x(C-x) = Cx-x2
1
y =
l_/ X ~~ X
* * *
^at (Bernoulli family Swiss
To math never gave a miss.
And 'twas always a fight
!As to who had the right
To publish which, what, and this.

1 000 Chapter 34 DIFFERENTIAL EQUATIONS
34.3 Formulating Differential Equations
Phenomena and processes in science and technology may
sometimes develop by discrete steps but do so more often in a
continuous manner, and then require differential equations
for their description and mathematical analysis. The
formulation of these differential equations requires physical
observations to be put into mathematical form, which is generally
not a very formidable undertaking.
These differential equations are of varied kinds and so are the
ways of solving them. Some equations, however, will resist
algebraic solution by purely mathematical procedures, but
may always be solved approximately by numerical or
graphical methods.
Population Growth
In biology, population is defined as the number of individuals
or organisms living in a certain area or, especially in botany,
the total mass of the species being considered.
A culture of bacteria of a given species "multiplies by
division" at a known rate of growth that is proportional to the
number P of bacteria in the culture, which is a function of time,
dP C dP
df=9--P; -p-=Jgd*; kiP = C + qt;
%/
P = Po-e^t ( where Pq = ec = initial population),
which is the general solution of the equation of growth.
The coefficient q can be determined experimentally by
photographing the culture through a microscope and counting the
number of bacteria that divide in a given space of time.
• At the beginning of a study, a culture contained 1000 bacteria of
a species which were found to split at an average rate of 1.2
times an hour. Assuming unchanged conditions, determine
the number of bacteria in the culture after 3 hours and after 24
hours.
Pit)
A
P (t) = number
of bacteria
t = time
We have p(3) = 1000.ei.23 ^ 36600
P (24) = 1000 • e12'24 * 3.2 • 1015 . •
There are several limiting growth-restricting factors - more
difficult to assess and to account for in a strictly mathematical
way - such as natural death and an accelerated demise of
bacteria through increased population density.

Section 34.3 Formulating Differential Equations
1001
Human population increase and forest growth are other
examples of the laws of exponential growth, which apply
equally for its inverse - exponential decay - as exemplified
by the dating of fossils by radioactive decay, deforestation by
acid rain, and the near-extinction of whole species of birds
and small mammals by the importune use of pesticides.
• From 1890 onward, an area has experienced a steady average
annual population growth of 0.3%, and in 1990 numbered
500 780 inhabitants.
- What was the population in 1890?
- Assuming a continued growth at the same rate, what will
the population be in 2090?
With 1990 as zero year, the population in the year t is P(t),
increasing at the annual rate of q = 0.3%,
— =a-P- P = Pc\ • e00°3*
d* q ' °
where Pq is the population in 1990, that is, Pq = 500 780 .
P (1890) = 500 780 • e~ 0003" 10° * 371 000
P (2090) = 500 780 • e0003 10° * 676 000 .
Human Population Explosion
In 1600 the Earth's human population was about 0.5 • 109; in
the second half of the 19th century it passed 1 • 109; in 1930 it
reached about 2 • 109, a figure that in 1995 was nearly
threefold, or 5.7 • 109.
In the above problems, we assumed unchanged conditions.
The reality of a continued human population growth is that the
Earth ultimately will reach a stage of insufficient resources to
support even more people, which would put a halt to the
population growth; persistent high birth rates would result in
generally shortened lives. The alternative would be humankind's
willingness to decrease reproduction to no more children than
to replace the parents; fewer births would result in increased
longevity.
Iteration and Chaos: p. 348 Predictions of population growths may suffer from chaotic
changes that result from sensitive dependence on initial
conditions used in the mathematical formula.
Radioactive Decay
Radioactive isotopes decay exponentially at a rate
characterized by their half-life, that is, the time required for the
intensity of the radioactive emission to diminish by half.
Radioactive decay is described mathematically by the same
differential equation as exponential growth, with the
difference that the proportionality factor is negative instead of
positive.

Chapter 34 DIFFERENTIAL EQUATIONS
Measurements of the radiation from a given isotope have
established that it diminishes by 2% in a one-year period;
calculate the half-life of the isotope.
If p is the decay constant, and R (t) the radiation at time t, we
= pR,
have jjj
d*
whose general solution is
R = CePt,
where C is the arbitrary constant.
To find C, use the information that at t = 0 the amount of the
radioactive element is R(0),
R(0) = Ce°; C = R(0) .
To find p, note that when t = 1 year, there is 98% left of the
original amount R (0),
0.982? (0) = R(0)eP
In e^ = In 0.98
p In e = In 0.98 ; p = In 0.98 .
With the half-life x years, we find
0.5 #(0) =R(0) ePx
and, inserting the value of p,
0.5 #(0) = R (0) e*ln °-98
In 0.5 = x ln 0.98 ln e ; x & 34.3 years. •
Radiocarbon Dating
The radioactive carbon isotope 14C is continually formed in the
Earth's atmosphere. Present in carbon dioxide, 14C is absorbed
from the air by plants and passed on to animals through the
food chain, resulting in the same proportion of 14C in the
carbon contents of living organisms as in the carbon reservoir
of the atmosphere. When an organism dies, it ceases to absorb
14C, whose proportion of the total amount of carbon then
steadily decreases. The half-life of 14C is 5730 ± 40 years.
Developed by the American physicist Willard F. Libby about
1946, the radiocarbon method is widely used to date 500- to
50 000-year-old fossils and archaeological finds. By
measuring the amount of residual 14C of a dead organism and
comparing the amount with the content of a living organism, the
death date can be estimated, or the age determined of plant and
animal products (papyrus, paper, cloth, charcoal, hides, etc.).
Estimate the age of a quiver which has 77.7% of its original
14C, whose half-life is 5730 ± 40 years.
If p is the decay constant, and R (t) the radiation at time t, we
have dR
-TT = pR\ R = CeP*,
at
where C is the arbitrary constant.

Section 34.3 Formulating Differential Equations
1003
R(0) = CeP°; C = R(0) .
With the half-life 5730 ± 40 years,
0.5 R(0) =#(0)e^5730±4°)
In 0.5
P " 5730 + 40 *
When t = x years, there is 77.7% left of the original amount
R(0),
0.111R(0) = R(0)ePx
1 nnnn In 0.777
p x In e = In 0.777 ; x = .
P
(5730 ± 40) ♦ In 0.777 „_,-_
x = ;—— = 2086 ± 15 years.
Compound Interest
Capital invested with interest may be compounded at any
agreed interval, e.g., annually, quarterly, daily, or
continuously.
$100 is deposited at a 6% annual interest rate, compounded
continuously; find the total value of the investment at the end
of 10 years.
The principal S grows at the rate given by
as
6t
= 0.06 S;
~ dS = J 0.06 d*
InS = 0.06* + Ci
S = e0.06^ + C1 = e0.06qeC1]>
and, as eci may be written C, the general solution is
S = Ce006' .
The initial condition gives
S(0) = 100 = Ce006' ° ; C = 100 .
At the end of 10 years, the principal will be
£(10) = 100 e006*10 = $182.21. •
If compounded annually at 6% annual interest rate, the
principal of the deposited $100 would be
S = 100(1 + 0.06)10 = $179.08;
quarterly,
S = 100
(^ 0.06 V10
V 4
= $181.40;
thus - to the surprise of many a disgruntled saver - there is no
great difference in the principal obtained with continuously
compounded interest.

1004
Chapter 34 DIFFERENTIAL EQUATIONS
Continuous Dilution
• A tank contains 10 kg of sodium chloride (NaCl) in 1000 liters
of water, which is being continuously diluted by 5 1/min of
fresh water being added to it during active mixing by means of
a rotary impeller, at the same time forcing an equal amount of
the solution out of the tank.
The NaCl concentration of the solution is originally 10 g/1,
that is, 10 kg in 1000 liters; how many minutes will be
required for the concentration to drop to 3.2 g/1 ?
5 liters I minute
5 liters I minute
The drainpipe removes 5 liters of solution every minute,
corresponding to 0.005 of the total volume. Thus the concentration
obeys the law
c(t) = 10 • e-0005';
for the sought time T, we have
3.2 = 10 • e-0005T ;
In 0.32
T =
0.005
= 200 I In 0.32 I * 228 min = 3h48min .
A 6-m3 tank contains 4.20% C02 (that is, 4.20% of the total
number of molecules in the gas mixture is CO2). Air with a
CO2 content of 0.03% is led into the tank at the rate of
2 m3/minute and thoroughly mixed with the gas in the tank.
The mixture leaves the tank at the same rate.
In a mixture of gases, each gas expands to fill the entire
volume available to the mixture; thus, every individual gas
occupies a volume equal to the entire mixture.
What is the concentration of CO2 in the tank after 10 minutes?
Let each m3 of the gas mixture contain Q molecules.
The number of CO2 molecules in the tank at t minutes is N(t);
at£(0),
N(0) = 6 • 0.0420 • Q = 0.2520 • Q .
CO2 enters the tank at the rate of
2 • 0.0003 • Q = 0.0006 • Q molecules/min,
and gas mixture leaves the tank at the rate of
2Q
N(t) N(t)
6- Q " 3
molecules/min.

Section 34.3 Formulating Differential Equations 1005
/,997 N = °0^6; Q • [e'd'/3 cU= 0.0018-Q+Ce-"3.
„. d t/ 3 J
dN
The concentration of COq in the tank changes at the rate-r~r
a t
dN N dN N
-^- = 0.0006 • Q - — ; -^- + — = 0.0006 . Q,
with the general solution
0.0006 • Q
J dt/3
Since N (0) = 0.2520 • Q, we have
0.2520 • Q = 0.0018 Q + Ce°; C = 0.2502 . Q,
and at t = 10,
JV (10) = 0.0018 • Q + 0.2502 • Q • e" 10/3
^ 0.0107 • Q molecules of C02
and the concentration
0.0107 Q
6Q
100 ^ 0.18% C02
Cooling and Heating
The rate of change of temperature of an object is proportional to
the temperature difference between the object and the ambient
medium. To enhance the concept of differential equations, all
other physical laws are ignored in the following two problems.
A turkey is taken from the refrigerator at 2° C and placed in an
oven preheated to 200° C and kept at that temperature; after
30 minutes the internal temperature of the turkey has risen to
16° C. The fowl is ready to be taken out when its internal
temperature reaches 88° C.
Determine the cooking time required.
The temperature 0 of the turkey increases at the rate d 61 dt,
d0
-£= c(2OO-0);
dO f J
= c \ dt,
200 -6
with the general solution
c -t = Ci - In (200 - 0),
where we insert the given initial and intermediate conditions:
t 0 200-0
(min.) (°C) (°C)
0 2 198 Cx = In 198
30
30 16 184 c = — (In 198 - In 184)
T 88 112
and obtain
^ «~ In 198-In 112 ^^ . nu rnmin A.
T = 30 • :—77^—:—tttt = 233 mm = 3n53min * 4 hours.
In 198 - In 184

Chapter 34 DIFFERENTIAL EQUATIONS
A slice is cut from a loaf of rye bread fresh from the oven at
180° C and placed in a room with a constant temperature of
20° C. After 1 minute, the temperature of the slice is 140° C.
When has the slice of bread cooled to 32° C?
The temperature 9 of the bread decreases at a rate proportional
to the temperature difference between it and the room air, that
is,
Tj2=c(e-20) => ^=-c«9-20);
0-20
dO = -c jdt,
with the general solution
c -t = Ci -In (0-20),
where we insert the given and intermediate conditions:
t
(min.
0
1
T
ooling
9
) (°C)
180
140
32
time is
In 16C
In 160
0-20
(°C)
160 Ci
120 c
12
) - In 12
- In 120 *
= In 160
= In 160 -
9 min.
-In 120
The Proverbial Bathtub
For a long time, maybe since the days of Archimedes, bathtub
problems have appeared in mathematical textbooks. Here is
ours.
• Mr. Crackpot's bathtub leaks and loses water at a rate that is
proportional to the volume of the water in the tub. With the tap
open all the way, he continuously adds 5 1/min into the tub,
which will just balance the loss through the drain when the tub
contains 105 liters of water.
One morning, arising late but still wanting his bath, he could
only allow the water to run for 15 minutes before he got into the
tub. Determine the water volume at the beginning of the bath.

Section 34.3 Formulating Differential Equations
1007
Solution:
Let the volume in the tub be v(t), the water supply rate q\, and
the drain rate q2 ,
q2 = c -v(t),
where c is determined by the equilibrium conditions,
Ql = Q2', 5 = c-105; c = — .
We then have the differential equation
di;
v
91-92 =dT=5-2i'
and the general solution
p. 997
v = 5 •
i
Jd*/21
$dt/21
dt oi . p^/21 . fi
where the initial condition v (0) = 0 gives C2 = - 105 , and
i;(15) = 105(l-e-15/21) ^ 54 [liters] .
Fall through a Resisting Medium
*><V^ L.
sx-g&
The acceleration a of a moving body is directly proportional
to the force F that acts on it and inversely proportional to its
mass /n,
F = kma,
where the magnitude of the coefficient k depends on the units
used. With SI units (mass in kilograms, acceleration in
meters per second squared, and force in newtons), the
coefficient k becomes 1,
IN = 1 kg • 1 m- s-2 .
The forces acting on a body in free fall through the atmosphere
are its weight, which is the product of mass m and the
acceleration of gravity g = 9.81 m- s-2, and the air resistance, which
may be considered to be proportional to the square of the
velocity of the fall.
As acceleration is the first derivative of velocity, we can set up
the equation
di;
F = m
dt
= m- g - k • v
where the coefficient k can be approximated by 0.3 with SI units
and in the relevant velocity region.
An object of mass 12 kg falls freely through the atmosphere.
Acceleration of gravity is 9.8 m/s2.
- If the initial velocity is zero, what is the velocity after
15 seconds?
What is the maximum velocity attainable?

1008
Chapter 34 DIFFERENTIAL EQUATIONS
p. 1007
Solution:
The equation describing the fall is
dv
12 -77- = 12 • 9.8 - 0.3 v2
at
or
dv_
dt
+ 0.025 v2 = 9.8
With the integrating factor e ' °-025dt = e0025 ', we obtain
1
1,2 =
,0.025 t
J9.8e°-025' d*
= Xe-°025^ + 392.
The initial condition t = 0; v = 0 gives K = - 392 ;
At £ = 15, we have
u2 = -392 e"002515 +392 * 123 [m/s]2; i; * 11.07 m/s .
At maximum attainable velocity, the acceleration is 0; thus,
■j— = 0. Solving for v in
F = 12-7-= 0 = 12 • 9.8 - 0.3 v2
dt
we find the maximum velocity ^|S92 m/s?* 19.80 m/s.
Natural Oscillations
Free Oscillations
Oscillating Motion
Natural oscillations - or free oscillations - develop in a
mechanical system whose equilibrium has been disturbed,
when the system returns to normal without interference from
outside forces.
A simple illustration of free oscillations is offered by a
mechanical spring at whose free end is placed a concentrated
mass m which causes an extension d of the spring beyond its
natural length I to a point of equilibrium p, where the weight of
the mass is balanced by the tension force in the spring.
////////////////////
J
equilibrium level of
d
±
mass - spring system
Harmonic Oscillations
If the mass is pulled down a distance sq below the equilibrium
level, the additional force set up in the spring is proportional
by a factor k, the spring constant, to the elongation sq of the
spring.
When the mass is released, the system will attempt to return to
its equilibrium position by harmonic oscillations with the
amplitude sq.

Section 34.3 Formulating Differential Equations
1009
Since force is proportional to the product of mass and
acceleration, which is the second derivative of distance with
respect to time, we have
m s = - k s ,
the negative sign signifying a reaction force; the "dotted"
derivative s' denotes the second derivative of distance with
respect to time.
By rearrangement, we have the fundamental equation of an
undamped harmonic oscillation,
s + — • s = 0,
m
where s denotes the second derivative of distance with respect
to time.
equilibrium level of
-► t
mass - spring system
If the surrounding medium presents resistance to the motion,
the equation must be supplemented by a term representing the
attenuating force, which is often proportional to the velocity of
the oscillating body, that is, to the first time derivative of
distance.
We then have the fundamental equation of a damped
harmonic oscillation,
C . K
s + — • s + — • s = 0 .
m m
cilLii
equilibrium level of
mass - spring system
A body whose mass is 1/2 kg pulls a spring down 1/5 m and
gains equilibrium. The body is then pushed 1/ 10 m above
the equilibrium position and released. A damping force
numerically equal to the velocity is present. Acceleration of
gravity is 10 m/s2.

1010
Chapter 34 DIFFERENTIAL EQUATIONS
Solved on p. 1021
S=U-
equiHbrmjn position of
spring & mass
Find the differential equation describing the position of the
body as a function of time; the mass of the body is assumed to
be concentrated at the end of the spring.
Solution:
To determine the spring constant k, we have
i (kg) x 10 (m/s2) = \ k (m); k = 25 (N/m).
Using the value of k and the information that the damping
force is numerically equal to the velocity, we may form the
differential equation
or
0.5 s + s + 255 = 0
s + 2 s + 50 s = 0.
The particular solution is obtained from the initial values
Forced Oscillations
and
distance 5 = - -tt meter at £=0
velocity s = 0 at t = 0.
Damped oscillations eventually die out; forced oscillations
continue for as long as the external activating force remains
coupled to the system.
Forced oscillations develop in a system regularly activated by
external forces (e.g., the pendulum of a clock, a swing pushed
on the downswing).
The fundamental equation of a forced oscillation has the form
s + — • s + — - s = Fit).
m m
where F(t) may be any periodic function.
equilibrium level of
mass - spring system

Section 34.3 Formulating Differential Equations
1011
The differential equation
— s + 4 s + 30 s = 3 cos 51,
3
with initial values s(0) = -r and s (0) = 0, may represent a
system composed of a mass of 1/3 unit, which is attached to a
spring whose spring constant is 30 units, with a damping force
numerically equal to 4 times the velocity and an external force
equal to 3 cos 51 units; the external force 3 cos 51 is, of course,
periodic. The initial values indicate that the mass is pulled
down 1/2 length unit from the equilibrium position and
released with an initial velocity of 0:
////////
__ « equ^Hbriumjwsition of |
c=4l±
s = 0
spring & mass
s=l/2
at £=0
T
fc = 30
3 cos 5£
Generality of a Mathematical Model
Many physical and technical phenomena or processes may be
described by the same mathematical model. Thus, equivalent
mathematical descriptions exist between units of mechanics
and, e.g., units of acoustics, hydraulics, gas flow, and fluid
flow.
As an example, we note the following analogies between
mechanical and electrical quantities:
Mechanical
Quantity
Electrical
Quantity
mass (m)
position (y)
f
velocity v =
V
d*
friction coefficient (/i)
[damping constant (c)]
spring constant (k)
external force [F(t)]
inductance (L)
electric charge (Q)
electric current (I)
resistance (R)
l/capacitance (l/C)
[capacitance (C)]
electromotive force (E)

1012
Chapter 34 DIFFERENTIAL EQUATIONS
p. 1010
By the differential equation for forced oscillatory motion,
d2v dv
and the above analogies, we have the differential equation
d2Q dQ Q
or
dl Q
L-^ + RI +f =E(t)
m^
R-
E(t)
Orthogonal Trajectories
A curve which intersects all curves of a given family at the
same angles is referred to as a trajectory; if the intersection is
at a right angle, we have an orthogonal trajectory.
The concept of orthogonal trajectories is used in many
branches of physics, e.g., electrostatics (lines of force due
to an electric charge and lines of constant potential);
thermodynamics (isothermals and heat flow lines); hydrodynamics
(streamlines and lines of constant velocity).
The task of finding equations of orthogonal trajectories
involves interesting applications of fundamental principles of
analytic geometry, differential and integral calculus, and
differential equations.
The family of circles, x2 + y2 = r2, and the family of straight
lines, y = m x, are examples of families of curves that are
orthogonal trajectories of each other:
► x

Section 34.3 Formulating Differential Equations
1013
The slope of the family of straight lines is defined by
(1) 771 = ~ .
X
Since the derivative may be interpreted as the slope of a curve,
we also have
(2) £=».
Combining (1) and (2), we have
(3) *2L =1
dx x
p. 555 The slopes of orthogonal lines are negative reciprocals.
Consequently, the slope of the orthogonal trajectory line of (1)
is the negative reciprocal of (3),
dv x
(4) T1 = "-•
dx y
Solving (4), by separation of variables, we obtain
J y dy = -\ x dx
1
2
which we rearrange as
-y2 = ~2 *2 + Ci
~2~=Cl
or
y2 +X2 = C .
Replacing C with r2 gives
y2 + x2 = r2,
which, as expected, is a family of circles with centers at the
origin of the orthogonal coordinate system.
The orthogonal trajectories of the following example are not so
easy to predict as the one in the introductory example.
Describe the family of orthogonal trajectories of
y =px2 .
To find the slope of the curves represented by the given
equation, we differentiate
dx-=2px'
y
Eliminate p by inserting p = — .
xl
Thus,
dy_ y_x _ 2y
dx x2 x

Chapter 34 DIFFERENTIAL EQUATIONS
Introducing the negative reciprocal of the right member of the
above equation, we find
dy x
dx = ~2y"'
which represents the slope of the orthogonal trajectory of the
given curve.
Solving the differential equation, we obtain
\ 2y dy = -J x dx
y2 = ~^x2 + C,
where C is an arbitrary constant.
Replace C byp and rearrange,
2 x2 y2 x2
which is a family of ellipses with centers at the origin of the
orthogonal coordinate system; they are orthogonal trajectories
of the family of parabolas y =p x2.
p = 0.25
p = 0.5
P = l
p = 2
► x

1015
34.4 Second-Order Ordinary Differential Equations
Directly Integrable Equations
A differential equation of the type
d2* « *
can be solved by direct integration of both sides.
• We have the equation
d2y
-7—ir = COS x .
dx*
By integration, we find, first,
dy
dx
and then
= J cos x dx = sin x + C\
y = f (sin x + C{) dx = - cos x + C\ x + C<i,
where C\ and C2 are arbitrary constants.
Find the particular solution of
d2y
X
dx2
ify(l) = 2 and/(1) = 1.
x2 = 1
d2y ,
Rewrite the equation -r-j- = x + x~l and integrate.
The first integration gives
dv r X
j— = J (x + x_1) dx = — + Inx + Ci
where Ci is determined by the condition y' (1) = 1,
y'(l) = l=y+lnl + Ci; Cx = ± .
The second integration gives
y =
j
^ (x 1 1 X X
r- + In x +— \ dx = -,r + xkix -7?+ C2 ;
V
C2 is determined by the condition
y(l)=|Ullnl-|+C2=2; C2 = | .
The particular solution is
1
y
= — (x3 - 3 x + 14) + x In x .

1016
Chapter 3 4 DIFFERENTIAL EQUATIONS
Linear Differential Equations
ARBOGAST Louis
(1759-1803)
The general linear differential equation of order n is
dny dn~^y dy
where an (x), an_\{x) ... ao(x), and f(x) are given functions of
x, or constants.
In terms of the differential operator
d
D =
dx '
introduced by the French lawyer and mathematician Louis
Arbogast in his Du calcul des derivations (1800), the equation
becomes
an(x)Dny + an_1(x)Dn-1y + ...+a1(x)Dy + a0(x)y = f(x)
or, simplified, n
^ai(x) Dly = f(x) .
i = 0
DU CALCUL
DES
DfiR IVATI ONS;
PAR L F. A. ARBOGAST,
De lliutitut national de France , Profeueur de
Maihematiquej a Strasbourg.
A STRASBOURG,
OS L'lMFftlMSMS OS HTUULT, fhthli.
▲ K Tin (i8oo)
Homogeneous
Nonhomogeneous
If f(x) = 0, the linear differential equation is known as
homogeneous; iff(x) * 0, it is nonhomogeneous (or inhomoge-
neous).
This use of the term homogeneous has nothing in common
with its use to denote differential equations consisting exclu-

Section 34.4 Second-Order Ordinary Differential Equations
1017
Linear Dependence
Linear Independence
sively of homogeneous functions of the same degree in
x andy. To avoid unnecessary confusion, we will specify
linear when referring to the types of equation considered here.
Linear dependence and independence are concepts of
importance in the study of homogeneous and nonhomogeneous
linear differential equations. A collection of functions is
linearly dependent if one of them can be expressed as a sum of
constant multiples of the others; if this is not possible, the
collection is said to be linearly independent.
y\{x) = 3x andy2(x) = x are linearly dependent, one being a
multiple of the other; y\{x) = 3x and y$(x) = x2, on the other
hand, are linearly independent.
Homogeneous Linear Equations
Ifyi>y2 ••• Jn are linearly independent solutions of a
homogeneous linear differential equation of order n,
/ x &ny / x d^V , . dy
a„ (x) +an_i(x) 7- + ... +ai (x) -7— +ar\(x)y = 0
dxn dx"-1 dx uv
or
n
Y,ai(x)Diy = 0 ,
i = 0
then the general solution is
n
y= ^,ciyi = Ciyi + C2y2+'~ + Cnyn ,
i = o
where all C\, C2 ..., C^ are arbitrary constants.
Reduced Equation
Nonhomogeneous Linear Equations
A nonhomogeneous linear differential equation of order n9
n
2^ai (x) Dly = f(x) ,
i = 0
differs from its homogeneous counterpart, often called the
reduced equation in that the right-hand member of the
equation is * 0.
The general solution of a nonhomogeneous linear
differential equation is the sum of
any of its particular solutions, and
a complementary solution,
the latter of which is the
general solution of the reduced equation.

1018 Chapter 3 4 DIFFERENTIAL EQUATIONS
Homogeneous Constant-Coefficient Linear
Differential Equations
The Auxiliary Equation
The method of solution to be described is applicable to higher-
order homogeneous linear differential equations of the type
n
^ai(x)Diy = 0 ,
; = o
whose coefficients are constants, but we will deal here
specifically with second-order equations,
d2y dy
Attempting y = erx as a general solution, we have
(a2 r2 + air + clq) erx = 0 ;
aserx cannot be 0, we see thaty = erx is a solution of the given
differential equation, if
a2 r2 + a\ r + oq = 0 .
Auxiliary Equation This auxiliary equation, or characteristic equation, has the
Characteristic Equation roots
a\ + V«i2 - 4 a,Q a,2 -a\ - \ai2 - 4
rl = 7TZ "> r2 =
«0«2
2a2 ' * 2a2
yx = erix and y2 = erzx will then be particular solutions of the
differential equation.
p. 311 Depending upon the value of the discriminant
D = ai2-4a0a2,
we distinguish three cases:
D > 0 : ri, T2 real and distinct (unequal)
D = 0 : ri, T2 real and equal
D < 0 : r*i, r2 complex conjugates
Real and Distinct r\ and r2
If the discriminant is positive, the roots r\ and r2 of the
auxiliary equation
a2r* + air + ao = 0
are real and distinct and the general solution of the
homogeneous linear differential equation,
a2 y" + ai y'+aoy = 0,
is
y = deri* + C2er2* ,
where C\ and C2 are arbitrary constants.

Section 34.4 Second-Order Ordinary Differential Equations 1019
Real and Equal n and r2
If the discriminant is 0, the roots of the auxiliary equation are
real and equal,
n = r2 = r;
the formula above gives
y1 = C1erx + C2erx = C3erx ,
which contains only one arbitrary constant and therefore does
not qualify as a general solution. To get around this
difficulty, we consider
y2 = x erx
as a possible second solution, and verify that it is.
d2 d
a2 • —7" [x - erx] + a\ • -j— [x • erx] + an • x • erx
dxz &x
= a2 (2 r erx + r2 x • erx) + a\ (erx + rx erx) + an • x • erx
= erx (2 a2 r + a{) + x erx (a2 r2 + a\ r + an).
Both expressions (2 a2 r + a{) and (a2 r2 + a\ r + an) are = 0 when
a\
r = --— , that is, when
Za2
ai2 -£ana2 = 0 .
Since yY = erx and y2 = x erx are linearly independent, the
general solution of the homogeneous linear differential
equation
a2yn + aiy' +a0y = 0
is
y = Cierx + C2xerx,
where C\ and C2 are arbitrary constants.
Complex Conjugate r\ and r2
If the discriminant is negative, the roots of the auxiliary
equation (a2 r2 + a\ r + an = 0) are the conjugate complex numbers,
-a\
n r2 = -5— ± 1
"y ai2-4a0a2
2 a2 ~ 2a2
or
ri = a + bi; r2 = a-bi,
where
a= --5— ; b=-z— \ Uiz-4a0a2 .
Z a2 ZCL2 ^ ' '
The general solution becomes
y = de(a + bi)x + C2e(a-bi) x
= ea*(Ciei6* + C2e-i6*).
Equations having solutions of this type occur primarily in
oscillation problems.

1020 Chapter34 DIFFERENTIAL EQUATIONS
To obtain a solution without imaginary or complex number
p. 792 exponents, we use Eider's formula
elx = cosx + i sinx ,
and the corollaries
eibx _ cos 6x+ i sin6x ; e~lbx = cos bx-i sinbx ,
which give
y = eax (Ai cos b x + A2 sin b x) ,
where
Ai = Ci + C2; A2 = i(Ci-C2).
• Find the general solution of
y" + y' -2y = 0
and the particular solution for y (0) = 1; y' (0) = 0 .
The auxiliary equation
r2 + r - 2 = 0 ; (r - 1) (r + 2) = 0
has the roots
1=1; r2 = -2.
Inserting those values in y = C\ erix + C2 er2x, we obtain the
general solution
y = C1ex + C2e-2x,
where C\ and C2 are arbitrary constants.
Differentiating y gives
y ' = Cie*-2C2e-2*.
y (0) = 1 and y ' (0) = 0 require that
Cie° + C2e-° = 1 1 _2 _1
Cie°-2C2e-° = 0 J Cl = 3 ; °2 = 3'
and the particular solution is
y = -• e* + -- e~2x .
• Solve the equation yu +y' +y = 0 .
The auxiliary equation r2 + r + 1 = 0 has the roots
1 A .V3
r*r2 = -2 ± !— •
Replacing in
y - eax (j^1 cos b x + A2 sin b x) ,
we find the solution
x/2(a ^ A • ^ ^
y = e_x/z I Ai cos-t-x + A2 sin-t-x I,
where Ai and A2 are arbitrary constants.

Section 34.4 Second-Order Ordinary Differential Equations
1021
y" + 2y' + 50y = 0is the equation of a damped oscillation.
Find the particular solution for y (0) = - — and y ' (0) = 0 .
The auxiliary equation
r2 + 2r + 50 = 0
has the roots
ri, r2 = -1 ± 7 i.
The general solution is
y = e~* (Ai cos 7 t + A2 sin 7 t)
Since y (0) = - —, we have A\ = - — ;
y = e~* (A2 sin 7 £ - — cos 71)
/
1\
LV
7A* + u
cos 7 £
/
J
7\
Ao- —
K
10
sin It
J
,-t
Since y ' (0) = 0, we have
7A2 + -
• 1
r
7\
A2 + 10,
• 0
1 = 0; A2 = - —
1_
70
Hence, the particular solution is
y(t)
_ c-t
( 1 1^
- -=r sin 7 £ - — cos 7 £
V «0 10 y
y = /(0 = e-' (-^ sin 7* -i cos 7*)

1022 Chapter34 DIFFERENTIAL EQUATIONS
Harmonic Differential Equations
Equations of the form
d2y
+ n2y = 0,
dx2
where n is any number, are called simple harmonic
differential equations, and occur in problems involving periodic
motion described by sine or cosine functions of time.
Its auxiliary equation, r2 + 0 r + n2 = 0, has the roots r = 0 ± i n.
We find the solution
y _ e0x (^ cos nx +A2 sin n x) = A\ cos n x + A 2 sin n x,
where A1 and A<i are arbitrary constants.
d2y
Similarly, for —r- - n2y = 0, we find r = ±n .
dx2.
Consequently, yx = enx and y2 = e~nx are two linearly
independent solutions, and we obtain the solution
y = C1enx+C2e~nx .
p. 536 The above solution may be expressed in terms of hyperbolic
functions:
cosh nx =
r\TtX 1 cx—flX
2
enx - cosh nx + sinh n x
enx _ e-nx J e- nx _ cosh nx - sinh n x
sinh n x =
We may now write
y = (Ci + C2) cosh n x + (Ci - C2) sinh n x,
or
y = Ai cosh n x +A2 sinh n x .
Find the particular solution of
d2y
dx2
- 4y = 0
for y(0) = 1 andy' (0) = 5, expressed in exponential functions
and in hyperbolic functions.
Rewriting the equation
d2y
-22y = 0,
dx2
the general solution is
y = C1e2x+C2e~2x
with
yl = 2C1e2x-2C2e~2x .

Section 34.4 Second-Order Ordinary Differential Equations 1023
y (0) = 1 and y ' (0) = 5 give
Cie°+C2e°=ll Ci + C2 = l 1 Cx = 7/4
2Cie°-2C2e° = 5 J Ci-C2 = 5/2J C2 = -3/4
Thus, the particular solution in exponential functions is
y = — e2*- — e~2* .
J 4 4
We have
e2* = cosh 2 x + sinh 2 x
e-2* _ Cosh 2x-sinh2x ;
7 7 3 3
y = t cosh2x +— sinh 2 x - -r cosh2x + — sinh 2 x
J 4 4 4 4
5
y = cosh 2x+r sinh 2 jc .
In the differential equation
d2y k
-4-+ —y =o,
describing the undamped oscillatory motion of a spring-mass
system, m denotes the mass and k is the spring constant.
Find the general solution and the particular solution for
k = 18N/m, m= 2 kg andy (0) = - 0.05 m, y'(0) = 0 m/s.
We obtain the general solution
y = A1 cos^y— t + A2 sin^y— t .
Inserting the values of k and m in the general solution,
18
T7- t
y = A1 cos"W— t + A2 sin^W-
= Ai cos 3 t + A2 sin 3 t.
y (0) = A1 cos 0 + A2 sin 0 ; A1 = - 0.05 .
So
y = - 0.05 cos 3 t + A2 sin 3 t;
y' = 0.15 sin 3 t + A2 3 cos 3 t;
y' (0) = 0.15 sin 0 + A2 3 cos 0 = 0 ; A2 = 0 ,
and the particular solution is
y = - 0.05 cos 3 t.

1024 Chapter 3 4 DIFFERENTIAL EQUATIONS
Nonhomogeneous Constant-Coefficient Linear
Differential Equations
Introduction
The general solution of a nonhomogeneous linear
differential equation of order n is
y = yP + C1y1 + C2y2 + '" + Cnyn,
where yp is some particular solution of the equation and
Qiyi + ^2 J2 + ♦♦♦ + Cn.yn istne complementary solution, that is,
the general solution of the corresponding homogeneous linear
differential equation.
We shall discuss a method of solving higher-order non-
homogeneous linear differential equations with constant
coefficients,
dny d/l_1y dy .. .
a"d^" +an-1 ~^TT + -+ai dl + a°y = fix)'
but restrict ourselves here specifically to second-order
equations,
d2y dy
0,2 d*2~ +aidx + a°y = ^X) '
d2y
A simple equation of this type, —— + y = x, has the reduced
j. dxz
equation
Complementary Solution with the solution - known as the complementary solution -
y = Ai cos x +A2 sin x ,
where A1 and A 2 are arbitrary constants.
Examination shows that yp = x is a particular solution, and we
obtain the general solution
y = A\ cos x +A2 sin x + x .
It is only rarely that a particular solution of a
nonhomogeneous constant-coefficient linear differential equation can
be found by mere inspection, as in the above case. We shall
therefore describe both the more general method of variation of
constants and the simpler method of undetermined
coefficients, which is adequate for special cases of nonhomogeneous
constant-coefficient linear differential equations, commonly
encountered in physics.

Section 34.4 Second-Order Ordinary Differential Equations 1025
Variation of Constants
The method of variation of constants, also called variation of
LAGRANGE Joseph Louis parameters, is due to Lagrange, French mathematician and
(1736-1813) physicist.
If the differential equation
d2y dy
(1) ^ dx^ + aidx + a°y = ^X)
has the complementary solution
(2) yc =C1y1 + C2y2
then a particular solution of the form
(3) yp =u(x)yi + v(x)y2
can be obtained by restricting the functions u(x) and v (x) to
satisfying the condition
(4) ux (x) yi + vx (x) y2 = 0.
Differentiating (3), we obtain
d r n
JJ" Wpi = u(x)y\ + y\ux(x) + v(x)y,2 + ^2 ^' to,
which by virtue of (4) may be reduced to
d r ,
(5) -^ \yp\ = u{x)y\ + v(x)y'2.
Differentiating (5), we find
d2
(6) —y yp = uy"i + y\ux + v y"2 + y'2i/.
Inserting (3), (5), and (6) in (1) leads to
a2(uyx\ + y'i ux + v y"2 + y'2 u')
+ ai(uyxi + vyx2) + ao(uyi + v y2) = f(x)
or, rearranged,
(7) u (a2y"i + aiy\ + a0yi) +v (a2y"2 + aiy'2 + a0y2)
+ a2(y\ux+ yx2v*) = f(x) .
Since, by definition, y\ and y2 are solutions of the
homogeneous equation, we have
«2y"l + «iy'l + «oyi = 0; a2y"2 + aiy'2 + ao;y2 = 0
and (7) becomes
(8) y\u% + y'2 i/' =^ ; «2*0.
a2
We now have the simultaneous equations (4) and (8)
yiu ' + y2vx = 0
, , , , fW
y\u + y2 v = —-
a2
which give

1026
Chapter 3 4 DIFFERENTIAL EQUATIONS
U =
V =
y2f(*)
«2 (yiy,2-y\y2)
yifW
«2 (yiy 2 -y'iy2)
By integration
(9)
(10)
u = -
V =
1_ "
a2 „
a2 „
y2fW
yiy 2 -y 1^2
yi/X*)
yiy'2 -y'iy2
dx
dx
which we insert in (3) and obtain
yP = y2
a>2
yif(x) , i f
; ; dx-yi —
yiy 2 - y iy2 «2 J
y2f(x)
yiy 2 -y 1^2
dx
which is a particular solution of
«2
dx2
dy
+ aidx + a°y = ^X)
whose complementary solution is yc = C\ y \ + C<i y<i.
As may be expected, solving a differential equation by the
method of variation of constants is often quite a formidable
task.
Solve the equation y" + Ay = esc 2 x .
The homogeneous equation y" + Ay = 0 gives the
complementary solution
yc = A\ cos 2x +A2 sin 2x.
The particular solution is (since a<i = 1)
yP = y2
yi -fix) ,
; ; cbc-yi
yiy 2 -y 1^2 ./
y2 •/*(*)
yiy 2 -y iy2
dx.
where f(x) = esc 2 x, and
yi = sin 2x
y2 = cos 2 x
yi = 2 - cos 2jc
y 2' = - 2 • sin 2 jc
yi • esc 2 jc = 1
cos 2 x
y2 • esc 2 x =
sin 2jc
yi *y2'"~yi*y2 = ~ 2- sin2 2 x - 2- cos2 2 x = -2;
thus,
yP =
sin 2x
cos jc _ cos 2 jc f _
—: cbc - — dx
sin x 2 J
= -? • sin 2 jc • In | sin x \ - —x- cos 2 jc .
The general solution is
y = Ai sin 2 jc + A<i cos 2 jc + — sin 2 jc • In | sin x | - — • x • cos 2 jc

Section 34.4 Second-Order Ordinary Differential Equations
1027
Undetermined Coefficients
Perturbation Functions By identifying some special cases of perturbation functions
fix) in nonhomogeneous constant-coefficient linear
differential equations,
d^y dy
«2 TT + «i jjj- + aoy = fix) ,
we may dispense with the often cumbersome method of
variation of constants in favor of the method of undetermined
coefficients.
The basic principle of this method is to assume a particular
solution yp that is similar to the perturbation function except
that its coefficients are so far undetermined, and to solve the
system for the coefficients.
Amenable to treatment by the method of undetermined
coefficients are cases where the perturbation function is
o a polynomial;
o an exponential function;
o a function of trigonometric sines or cosines;
o a function of hyperbolic sines or cosines;
o a product or sum of such functions.
Polynomial Perturbation Functions
If the perturbation function fix) is a polynomial of degree n,
the first step is to attempt an n-th degree polynomial
particular solution yp>
yp = Anxn+An_1xn-1 + ...+A1x + A0,
where An, An_i, ..., Ao are undetermined coefficients.
The next step is to insert yp and its relevant derivatives in the
given differential equation, then equate the coefficients of both
members of the equation and solve for An, An _ i,..., Aq .
Exponential Perturbation Functions
If the perturbation function is an exponential function fix) =
b ecx , a first attempt at a particular solution yp should be
yp =A - xk • ecx,
where ^4. is a constant to be determined by the integrating
conditions and k is the lowest power of x that will prevent
duplication of terms that are already part of the
complementary solution - the terms must be linearly independent.
If no roots of the auxiliary equation are equal to c, then k = 0,
and we have
yp=A -ecx .
The constant A is determined by inserting yp and its
derivatives in the given differential equation.

Chapter 34 DIFFERENTIAL EQUATIONS
Trigonometric and Hyperbolic Perturbation
Functions
If f(x) is equal to (b • sin ex) or (b • cos ex), where b and c are
constants, we assume a particular solution
yp = xk (A • cos c x + B • sin c x);
if f(x) is equal to (b • sinh ex) or (6 • cosh ex), we choose a
particular solution
yp = xk (A • cosh c jc + B • sinh c x),
where A and B are constants to be determined and k is again
the lowest power of x that prevents duplication of terms in the
complementary solution.
The chosen yp and its derivatives are inserted in the given
differential equation and the coefficients of the equation
members are equated.
Product-Type Perturbation Functions
If the perturbation function is a product of several functions of
the kinds that allow solving the equation by the method of
undetermined coefficients, a particular solution may be
obtained as a product of factors of particular solutions.
Sums of Functions
If
d2y dy
and yPl and yP2 are particular solutions of
«2 y" + «i y'+«oy = gib) and a2y" + ai yl + a0y = g2(x),
respectively, then
yP=yPl+yP2
is a particular solution of the given differential equation.
Problems
Find a particular solution of
y" + 2y'-4y = 4x2-8x + 12 .
Since the right-hand member of the equation is a second-
degree polynomial, the particular solution is of the form
yp = A2x2+Aix+A0.
Differentiation gives
yp' = 2A2*+Ai; V' = 2A2.
Inserting yp and its derivatives, we obtain
-4A2x2 + (4A2-4A1)jc + (2A2 + 2A1-4Ao) = 4x2-8x + 12 .

Section 34.4 Second-Order Ordinary Differential Equations
1029
Equating the coefficients gives
-4A2 = 4
4A2-4Ax =-8
2 + 2A!-4Ao = 12
A0 = -3
Ax = 1
A2 = -1
Hence, a particular solution is
yP
— —jC t jC — O .
Find the particular solution of
y"-3y' + 2y = 4x
fory(0) = 7; y ' (0) = 2 .
The auxiliary equation r2-3r + 2 = 0 has the roots r\ = 2 and
r2 = 1; consequently, the complementary solution yc of the
given equation is
yc = C!e2* + C2e*
and the particular solution is of the form
yp = Aix+Aq
with
yp' = Ai and yp" = 0 .
Inserting yp and its derivatives, we have
-3A1 + 2(A1x+A0) = 4x + 0
2A1x + (2A0-3A1) = 4x + 0
and
yp = 2x + 3.
The general solution of the given equation is
y = C1e*x + C2e* + 2x + 3,
where C\ and C2 are to be determined.
Ao = 3
Ax = 2
= 4
a
>
= 0
4
8
y (0) = 7 = Cxe° + C2e° + 2 • 0 + 3 ; Cx + C2
Differentiation gives
y' = 2Cie2* + C2e* + 2;
y '(0) = 2 = 2C!e0+C2e° + 2; 2(^ + (¾
Hence, the sought particular solution is
y = - 4 e2x + 8e* + 2:x; + 3.
Find the general solution of
y" + 4y' + 4y = 5 e3* .
Begin by finding the complementary solution yc. The
auxiliary equation r* + 4 r + 4 = 0 has the roots
n = r2 = -2;
thus the complementary solution is
yc = C\ e~2x + C2x e~2x
with the arbitrary constants C\ and C2.

1030 Chapter 34 DIFFERENTIAL EQUATIONS
To find a particular solution yp, assume
yp = Axk e3*.
In the expression yp = A xk ecx, c is not a root of the auxiliary
equation. Therefore, k = 0, and
yp = Ae3x.
Differentiation gives
yp% = 3Ae3* ; yp" = 9Ae3*.
Inserting yp and its derivatives, we obtain
9A e3* + 4(3Ae3*) + 4Ae3* = 5e3x ; A = \ ,
and
yP = -e
3jc
The general solution of the given equation is y = yc + yp , or
y = Cie~2x + C2X e~2x +
q3x
5 '
Find the general solution of
y" + 4y' + 4y = 5e~2x.
The complementary solution is
yc = Cie~2x + C2X e~2x,
where C\ and C<i are arbitrary constants.
To find a particular solution yp, assume
yp = Axke~2x.
The terms of the complementary solution have the factors
e-2x an(j x e-2x gy choosing k = 2, duplication of terms that
already appear in the complementary solution is prevented.
Therefore,
yp=Ax2 e~2x.
Differentiation gives
yp = 2A(x-x2)e~2x
yp" = 2A(l-4x + 2x2)e~2x
5
Inserting these in the given equation, we find A = — ,
and thus
2
yP = 9 x" e
and the general solution is
K /y»" fV~ ^X
y=yc+yP = Cie-2x + C2xe-2x + r

Section 34.4 Second-Order Ordinary Differential Equations 1031
Find the general solution of
y'' + 4 y = sin 2 x .
Solving the homogeneous equation y" + 4y = 0, we obtain the
complementary solution
yc = A\ cos 2 x + A2 sin 2 x ,
where A\ and A 2 are arbitrary constants.
To find a particular solution yp, assume
yp = xk (A cos 2 x + B sin 2 x).
In order to prevent duplication of terms let k = 1, which gives
yp = x (A cos 2x + B sin 2 x).
Differentiation gives
yp' = - 2 A x sin 2 jc + A cos 2 jc + 2 £ jc cos 2 x + B sin 2 x
yp" = -4Ax cos 2x-4A sin 2x-4Bx sin 2x + 4B cos 2 jc
Inserting yp andyp\ we have
4£ cos 2x-4A sin 2 jc = 0 cos 2x + sin 2 jc ,
and by equating the coefficients,
A = -- ; B = 0.
4
We now have -,
yp = —~7X cos 2 x
and the general solution is
y = yc + Jp = Ai cos 2x +A2 sin 2 jc -7 x cos 2 jc .
Find the general solution of
y,f -y = 3 sin x .
The auxiliary equation r2 - 1 = 0 has the roots ri, r<i = ± 1,
giving the complementary solution
yc = Ci e* + C2 e"*
with the arbitrary constants C\ and C2 .
To find a particular solution yp, assume
yp = xk (A cos x + B sin x).
There is no duplication of terms from the complementary
solution; therefore, k = 0, and we have
yp = A cos x + B sin x
yp = - A sin x + B cos x
yp" = -A cos x - B sin x
Inserting yp and yp" in the given equation, we find A = 0 and
B = - 3/2. Thus,
3 • 3 •
yp = 0 cos * - 0 sin # = -^ sin jc ,
and the general solution is
3
y = C\ ex + C2 e~x - — sin x .

1032
Chapter 3 4 DIFFERENTIAL EQUATIONS
Find the general solution of
y" + y = (x - 1) Sin X .
The auxiliary equation r2 + 1 = 0 of the reduced equation
y" + y = 0 gives r = ± i.
The complementary solution yc is
yc = Ai cos x +A2 sin x ,
where A\ and A2 are arbitrary constants.
Assume a particular solution yp of a form similar to the right
member of the given equation, except that the coefficients are
undetermined,
yp = (Ax + B)xk cosx + (Cx + D)xk sin x ,
where we let k = 1 to prevent duplication; thus,
yp = (Ax2 + B x)cosx + (Cx2 + Dx) sinx
ypx = cos x (2 Ax + B + Cx2+Dx)
+ sin x (- A x2 - B x + 2 C x + D) ^
yp" = cosx[-(Ax2 + Bx) + 4Cx + 2(A + D)]
+ sin x [- (C x2 + D x) - 4 A x - 2 (B - C)]
j
jp + yp = x 'sin x -sin *
= 4 C jc • cos jc + 2 (A + D) cos jc
- 4 A jc • sin x - 2 (B - C) sin x
and, by equating coefficients,
A=-i; 5 = |; C=0; Z> = \
We now have
/
yP =
—x* + —x Icos X +
V
jc sin x
( 1
0 x2 + —x 1 sin x
V 4
JC COS X
X* COS JC
4 2 4
The general solution of the given equation is y = yc + yp, or
y = Ai cos x +A2 sin jc +
jc cos jc jc^ cos jc x sin jc
— _ ■ + ■
Find the particular solution of
y"+ 2y'-3y = x e"3*
for y (0)= landy'(0) = 2.
y" + 2y'-3y = 0; ri=l; r2 = -3;
the complementary solution yc is
yc = Ci^ + Cse-3*,
where C\ and C2 are arbitrary constants.
The right-hand member of the given equation may be written
(x + 0) e~3 x and we therefore assume
yp = (Ax + B)xk e-3x .

Section 34.4 Second-Order Ordinary Differential Equations
1033
p. 1030
To prevent duplication, let k = 1,
yp = e~3 x (A x2 + B x)
V = e-3x(-3Ax2 + 2Ax-3Bx + B)
yp" = e"3* (9Ax2-12Ax +9Bx + 2A
~\
>
6B)
j
ypn + 2yp-3yp = e~3* (9Ax2-12Ax +9Bx + 2A-6B
-6Ax2+ 4Ax-6Bx + 2B
-3Ax2 -3Bx )
= e~^x(
8Ax
+ 2A-4B)
A=~8>B =
We now have
^2
yP =
A
_1_
"16 •
and the general solution of the given equation is y = yc + yp,
y = C\- e* + C2 • e~3 *
' /y* ^ /y* 1
i—- 0 X
~\
yx = Ci-e* -3 02-6-^
^3 x2 x_ _0
~ 16 " 16
>
V
8
y(0) =1
y'(0) = 2
Ci + C2 = 1
Ci - 3 C2 - -jg = 2
^
>
j
Ci =
c2 = -
y
1—0 ^c
81
64
17
64*
Thus, the particular solution is
81
17
1—«j X _
1 O X
y ~ 64 64 8 16
Solve the equation
y" + 4y' + 4y = 5e3*+5e~2* .
Particular solutions of
y" + 4y' + 4y = 5e3* ; y" + 4y' + 4y = 5 e~2x
are, respectively,
1 CiX
ypi = —
yP2 =
5 x2 e~2x
A particular solution yp of the given equation is
ryijX K /y*^ cl~ £iX
yP =~+ 2 ;
its complementary solution is
yc = C\ e~2x + C2X • e-2*,
and the general solution is y = yc + yp, or
y = ¢16-^ + 02^-6-^ + -+ 2
where Ci and C2 are arbitrary constants.

1 034 Chapter 34 DIFFERENTIAL EQUATIONS
Series Solution of Second-Order Ordinary
Differential Equations
Up till now, we have only discussed methods for solving
second- or higher-order linear differential equations which
are suitable for equations with constant coefficients.
Solutions for second-order linear differential equations of the
form
that is, linear homogeneous differential equations whose
coefficients P and Q are functions of x, can be found by power
expansion methods.
Assume the solution
oo
y = f(x) = C0x° + C1x1 + C2x2 + C3JC3 +...+ Cnxn = XV
/1 = 0
and substitute power series expansions of y and its derivatives
for the terms of the given equation.
The result will be a power series formula, which is useful
provided that the series meets the proper convergence criteria.
Find the general solution and the particular solution of
y" +xy' +y = 0
for the initial values y(0) = 5 andy' (0) = - 3 .
Assume the series solution
00
y = I cnxn.
n = 0
OO OO
- 2 .
n=0 n=0
y' = I n Cnxn - 1 ; y" = I n (n - 1) Cnxn ,
OO OO
Xy* = X X n Cnxn ~ 1 = X n Cnxn -
/i = 0 /i = 0
Insert in the given equation,
OO OO OO
X n (n - 1) Cnx n ~ 2 + X n Cnx n + X Cn*n = 0 ,
/i = 0 /1 = 0 /1 = 0
and rearrange,
OO OO
X n (n - 1) Cnx n ~ 2 = - X (n + 1) C„x» .
/i = 0 /1 = 0
The next step is to obtain equal powers of x in the series
expressions, so that we can equate the coefficients.

Section 34.4 Second-Order Ordinary Differential Equations
1035
Replacing n by (n + 2) in the series expression for y", we
obtain
oo
oo
Xre(7i-l)C„*"-2 = I(re +2)[(n + 2)-l]C„ + 2*[(n + 2)-21
n=0 n+2=0
OO
= X (rc + 2) (rc + 1) Cn + 2 x ».
ft = -2
The rearranged equation becomes
oo
oo
Z(n + 2)(n + l)Cn + 2xn = - X (n + 1) Cn x n ,
n = -2 n = 0
that is,
(n + 2)(n + l)Cn + 2 = -(n + l)Cn ,
from which
Cft + 2 = ~
n + 2
, where n = 0,1, 2, 3, ... .
Thus, we find the coefficients
Co = -
Ca=-
Ca = -
Co
2
4
0
2 4
2i_ - Co
6 "246
C<ln -
(-D*Cq
2 ■ 4 - 6 ... (2 n)
Co =
c* =
c7 =
3
£3
5
3 • 5
-Ci
3 5 7
C'2ft + 1 =
(-DnCi
3 .5-7 ... (2 71 + 1)
00
With the assumptiony = Y,C nxn, we now have the general
solution n = 0
y = Cq + C\x ~—xz -~^ + 2~J 3~~5 " '"
or, by gathering terms with like coefficients Cq and Ci,
respectively,
y = c0
XA
f ~2 r4
v ~ T + 2 4 ~ 2-4-6
+ ...
J
+ Ci
y (0) = 5 gives Cq = 5 .
' /y* O /y* O /y* I
* " T + 3 • 5 " 357
+ ...
J

1036
Chapter 34 DIFFERENTIAL EQUATIONS
Differentiating y yields
y'= c0
r
2x
4 jc3
V
°-—+^-T
6x5
2 46
+ Ci
f
1 -
3 x2 5x4
7x6
\
3 3-5 3-5-7
y ' (0) = -3 gives Ci = -3.
Inserting Cq and Ci, we obtain the particular solution
y = 5
/
1 - *r +
JC'
JC<
V
24 2 46
/
-3
X'
X'
X
X -
V
35 3 57
p. 277 The series converges for \x\ < 1 by the ratio test.
Alternative Answers:
Sometimes a series may be recognized as the expansion of an
elementary function. Indeed, for one of the series obtained in
the last example, we have
" 2~+ 24" 2 46
+ ...
= 1 +
r x2
r *-2\2
v
2 !
x
f X2\3
K
3 !
+ ... = e-*2'2
V
The second series that is part of the general solution is not
an elementary function; and the particular solution for the
initial values y(0) = 5 andy' (0) = - 3 may be written
y = 5e
,-*2/2
f
X'
x%
X
X -
\
35 3 57
If the initial values are specified, as here, for x = 0, the solution
oo
y - ^, C nxn is a suitable assumption. However, if the initial
/i = 0
values of x are prescribed at points x * 0, we must choose
another series expansion of y. The following example is a
case in point.
Find the particular solution of
y,,-2(x-l)y, + y = 0
for the initial values y (1) = 3 and y' (1) = - 2.
Since the initial conditions are specified at x = 1, it is
advisable to find a series solution in powers of (x - 1) instead
of powers of x. The advantage will be evident when solving for
the constants of the particular solution.

Section 34.4 Second-Order Ordinary Differential Equations
1037
With X = (x - 1), the given equation becomes
y" -2Xy' + y = 0.
Assume the solution
oo
y= £c„J?*
/i = 0
with
oo
oo
y = InCnX"-1; y" = In(n-l)CflX»
/i = 0 /1 = 0
-2
and
oo
oo
Xy> =X^nCnX"-1= ^nCnXn-
/i = 0 /1 = 0
To obtain equal powers of x in both series expressions replace n
with (n + 2) in the expansion of y"; thus, as in the previous
example,
oo
oo
I I
= Y,n(n-VCnXn-2= Y,^n + 2^n + 1^Cn+2Xn
/i = 0
/i = -2
The given equation may now be written
oo
oo
oo
Y,(n + 2)(n + l)Cn+2Xn- 2 £" CnXn + XC»X" =0
/i = -2
or
/i = 0
/1 = 0
oo
oo
£(" + 2)(71 + 1)0,^2^ - Y,^2n-^CnXn =0
/i = -2
/i = 0
or
oo
oo
Y,(n + 2)(n + l)Cn + 2Xn = ^{2n-l)CnXn .
/i = 0
/1 = 0
Equating the coefficients, we have
(n + 2)(n + l)Cn + 2 = (2n-l)Cn
and
(2n-l)
L'" + 2"(n + 2)(re + 1) °"
re = 0,1,2,3...
We find the coefficients
c2=-
3C2
43
7C4
6 • 5
11 C4
C
2 •
" 4
" 6
0
1
-3C0
•3-2-1
-7-3C0
-5-4-3-2
-11-7-
• 1
3C0
8-7 8-7-6-5-4-3-2-1
Cl + 2 =^3 =
3 • 2
CK =
Ci =
Cq =
e£c.
5C3
5 -4
9C5
7-6
13 C*
9-8 9-8-7-6-5-4-3-2
5
7
5Ci
4 3
- 2
9-5C!
■6-5
13
4 -
•9-
3 • 2
5CX

1038
Chapter 34 DIFFERENTIAL EQUATIONS
We can distinguish a pattern; thus:
Co =
Ca =
c« = -
c8 =
Co
2 !
3C0
4 !
3-7C0
6 !
3.7-llCo
8 !
n 3-7- 11... (4n-5) ^
C2» = " (27o1 c°
Cq =
CK =
C7 =
Co =
£1
3 !
5Ci
5 !
5»9Ci
7 !
5»9-13Cj
9 !
2/1 + 1 -
5 • 9 ... (4 n-3)
(2 71 + 1) !
n = 1, 2, 3
00
y =
£^CnXn gives the general solution
/1 = 0
y =
Coll -7T7*2 -£**-
+ c
^A + 3 ! A + 5 !
3 • 7 fi ^
6! - )
■x5 +^fx'
r
• • •
J
or, withX= x- 1,
y = c0
^y(x-l)2 - fy(x-l)4 - -y-(x-l)6
+ C
3 !
_5_
5 !
(x - 1) + TTT (x - 1)3 + TT (X - 1)5 + "^7T- 0* - 1)7 + ...
5 • 9
7 !
which for the given initial value y (1) = 3 gives Cq = 3
Differentiating y yields
yl =c0
o
2 !
(x-1)
3 -4
4 !
(x-1)3 -
+ C
1 + ^- (x-1)2 + ^(x-l)4 +
which for y' (1) = -2 gives C i = - 2 .
Inserting these numerical values into the general solution, we
now obtain the particular solution
y = 3
-^(x-l)2 - ^- (x-1)4
3 »7
6 !
(*-l)(
-2
(x-1) + -!y (x-1)3 +^j- (x-1)5 +^p (x-1)7 +

Section 34.4 Second-Order Ordinary Differential Equations
1039
cc/ still don't have all the answers, but Pm
beginning to ask the right questions"
Drawing by Lorenz; © 1984
The New Yorker Magazine, Inc.
The reach of differential equations is boundless and much
more can be said about methods for their solution. But not here.
With the foundation laid in this book, more advanced texts on
differential equations and other branches of mathematics
may now be the reader's oyster.

1041
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Titchmarsh, E. C: Mathematics for the General Reader (1959). New York: Dover, 1981.
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Yoshino, Y.: The Japanese Abacus Explained (1937). New York: Dover, 1963.

1048
Works Cited
The strength of those who have gone before should not be
gleaned solely from histories or random quotations. True
inspiration and power lie in the original works.
Numbers in bold face at the end of the entries refer to page number in this book.
* before a page number indicates display of a reproduction of the work.
Abel, Niels Henrik: Article in Journal fur die reine und angewandte Mathematik
("Crelle's Journal ", 1826); 300
Agnesei, Maria: Instituzione analitiche (1748); 680
: Analytical Institutions (1801); 680
Apollonius of Perga: Conies (c. 200 B.C.); 408
Arbogast, L.-F.-A.: Du calcul des derivations (1800); 686, *1016
Archimedes: On the Measurement of the Circle; 90
- : The Sand Reckoner; 29
Argand, Jean Robert : Les quantities imaginaires (1806); *88
Bamberger Rechenbuch (1483); *74, *118
Barrow, Isaac: Lectiones Opticae & Geometricae (1674); 675
- : Euclid's Elements (travel edition, 1686); *416
Bashkara: Lilavati (c. 1150); 299
: Vija-ganita (c. 1150); 299
Bernoulli, Jacob: Ars Conjectandi (1713); 30, *963
Bible, 1 Kings 7.23; 93
Bible, Mark 5.9; 14
Bible (1562 misprint), Matt. 5.9; 46
Bombelli, Raffaele: L Algebra (1572); 87
Boole, G.: Mathematical Analysis of Logic (1847); 217
- : An Investigation into the Laws of Thought, on which are founded the
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Briggs, H.: Logarithmorum chilias prima (1617); 152
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Btirgi, J.: Aritmetische und Geometrische Progress Tabulen (1620); 153
Buteo, Johannes: Opera geometrica (1559); 108
Cantor, G.: Publications in Acta Mathematica; 262
- : Uber eine Eigenschaft des Inbegriffes aller reellen Zahlen (1874); 257
- : "Beitrage zur Begrtindung der transfeniten Mengenlehre", Matematische
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Cardano, G.: Artis magna sive de regulis algebraicis liber unus (1545); *73, *299, 316, 322
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Carroll, Lewis: Through the Looking-Glass (1872); 227

BIBLIOGRAPHY: Works Cited 1049
Cauchy, A.-L.: Cours d'analyse (1821); *345
- : Legons sur le calcul differentiel (1829); *719
Cavalieri, Bonaventura: Geometria indivisibilibus continuorum; ^674
Ceulen, Ludolph van: Van den Circkel (1596); *92
- : Aritmetische en Geometrische fondamenten (1615); 92
Chu Shih-Chieh: Precious Mirror of the Four Elements (1303); *141
Codex Mendoza; 60
Copernicus: De revolutionibus; 464
Corpus Iuris Civilis; *40
Cotes, Roger: Harmonia mensurarum (1722); 581
Cremona, Gerard of: The Almagest (translation, c. 1175); 461, 463
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152
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Desargues, Girard: Broullion projet ... (1639); 434
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Fibonacci Association: The Fibonacci Quarterly] 288
Fincke, Thomas: Geometriae rotundi (1583); 465, 486
Fisher, R. A.: Statistical Methods for Research Workers (1925); *964
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reinen Denkens (1879); 217
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Gafurius, F.: Theorica Musice (1492); *922
Galilei, G.: Dialogue Concerning the Two Chief World Systems (1632); 257
- : Discorsi e Dimostrazione Matematiche (1638); ^893

1050 BIBLIOGRAPHY: Works Cited
Gauss, C. F.: Demonstratio nova theorematis omnem functionem
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Gibbs, J. W.: Vector Analysis (1881); 601
Grassmann, G.: Die lineale Ausdehnungslehre (1844); 601
Guldin, P.: Centrobaryca (1635 - 41); 866,884
Hamilton, W. R.: Lectures on Quaternions (1853); 601
Harriot, T.: Artis analyticae praxis (1631); 104
Hasper, W.: Handbuch der Buchdruckerkunst (1835); *39
Hay, R.-J.,: Traite de christallographie (1822); *399
Hawking, Stephen W.: A Brief History of Time; From the Big Bang to Black Holes (1988);
Hilbert, David: Grundlagen der Geometrie (1899); 369
Hobbes, T.: Leviathan; 3
H0eg, Peter: Smilla's Sense of Snow (1993); 70,370
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Hospital, G. F. de L': Analyse des infiniment petits (1696); ^679
Huilier, S. L': Principorum calculi differentialis er integralis (1795); 780
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Hutton, Charles: A Mathematical and Philosophical Dictionary (1796); 673
Huygens, C: De ratiociniis in ludo aleae (1656); 963
- : Horlogium Oscillatorium (1673); ^591
I Ching (c. 2000 B.C.); 184
Isidore of Seville: Etymologiae (c. A.D. 600 ); 5
Jode, C. de: De quadrante geometrico in quo quidquid (1594); ^490
Jones, William: Synopsis palmariorum mathesos (1706); 85
al-Kashi, J. M.: Risala al-muhitiyya ("Treatise on the Circumference") (1424); 91
Kepler, J.: Mysterium Cosmographicum (1596); ^398
: Astronomia Nova (1609); *398
- : Nova stereometria doliorum vinariorum (1613); ^-872
- : Harmonices mundi (1619); ^399
: Tabulae Rudolphinae (1627); 152
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- : Hisab al-jabr w'al-musqabalah (825); 100, 298
Kolmogorov, A. N.: Grundbegriffe der Wahrscheinlichkeitsrechnung; 965
Lagrange, J. L.: Theorie des fonctions analytiques (1797); ^686
- : Lecons sur le calcul des fonctions (1800); 686
Lambert, J. H.: Theorie der Parallellinien (1786); 369
Laplace, P. S.: Theorie analytique des probabilites (1812); 964
Leibniz, G. W. von: Dissertatio de arte combinatoria (1666); ^-217
: Acta Eruditorem (1694); 336
- : "Nova methodus pro maximis et minimis" in Acta Eruditorem (1694);
*676
L'Hospital, G. F. de, see Hospital, G. F. L'
L'Huilier, S., see Huilier, S. L'
Lincoln, A.: Gettysburg Address (1863); 21
Linne, C. von: Critica botanica (1737); 5
Listing, J. B.: Vorstudien zur Topologie (1848); 369

BIBLIOGRAPHY: Works Cited 1051
Luneschloss, J. de: Thesaurus mathematum reservatus per algebram novam; 296
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Menelaus: Sphaerica (c. A.D. 100); 463
Milne, A. A.: The House at Pooh Corner (1928); *303, 305, *946
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Napier, J.: Mirifici logarithmorum canonis descriptio (1614); 152
- : A Description of the Admirable Table of Logarithms (1616); *151
: Rabdologiae (1617); 174
- : Mirifici logarithmorum canonis constructio (1619); 152
Nehemiah: Textbook on Geometry (c. A.D. 150); 95
Neumann, John von, and Oskar Morgenstern: Theory of Games and Economic Behavior
(1944); 965
Newton, Isaac: Methodus fluxionum et serierum infinitorum (1671); 678
- : Philosophiae naturalis principia mathematica (1687),"Principia";
548, *895
- : "Tractatus de quadratura curvarum" in Opticks (1704); *677
: The Method of Fluxions and Infinite Series (1736); *678
Otho, Valentinus: Opus palatinum de triangulis (1596); 465
Oughtred, William: Clavis mathematicae (1631, 1647); 104,
- : Clavis mathematicae (3rd ed., 1652); ^109, 110
- : The Circle of Proportion and the Horizontal Instrument (1632); 177
: Canones sinuum (1657); 110
Pacioli, Luca: De divina proportioni (1509); *399, 418
Pao Chhi-Shou: Pi Nai Shan Fang Chi ("Pi Nai Mountain Hut Records"); 214
Pappus: Synagoge (c. AD. 340); 866
Pascal, B.: Essay pour les coniques (1640); 434
- : Traiti du triangle arithmethique (1665); ^141
Peano, Giuseppe: Arthmetics principia, nova methodo exposita (1899); 157
Pearson, K.: The Grammar of Science (1892); 964
Peet, T. E.: The Rhind Mathematical Papyrus (1923); 99
Penrose, L.S. and R. Penrose: "Impossible Objects: A Special Type of Visual Illusions";
British Journal of Psychology, Vol. 49 (1958); 374
Piltz, A.: Die gelehrte Welt des Mittelalters. Cologne: Vohlan Verlag, 1982; 218, 890
Pitiscus, B.: Trigonometria; sive de solutione triangularum tractatus brevis
etperspicuus (1595); 458,465
- : Trigonometria sive de dimensione triangulae (1600); 458, ^459, 465
: A Canon of Triangles (1614); 465
Poincare, J. H.: Analysis situs (1895); 369
Poisson, S.-D.: Recherches sur la probability de jugements (1837); 964
Ptolemy: Almagest (c. AD. 150); 367, 461, 463
: Syntaxis mathematica (c. A.D. 150); 463
Rahn, Johann Heinrich: Teutsche Algebra (1659); 105
Recorde, R.: Ground ofArtes (1542), 1558 ed.; *103, 118, 170
: Whetstone of Witte (1567); *107
Regiomontanus, J.: De triangulis omnimodis (c. 1464, in print 1533); *464
Reisch, Gregor: Margarita phylosophica (1446, 1503, 1583); *115, 116, *218, *463 *870,
*890, *922, *962

1052 BIBLIOGRAPHY: Works Cited
Rhaeticus, G. J: De lateribus et angulis triangulorum (1542); 464
- : Canon doctrinae triangulorum (1551); 464
Rhind Papyrus (c. 1650 B.C.); *98, 134, 297
Riemann, Bernhard: Uber die Hypothesen, welche der Geometrie zu Grunde liegen (1854);
383
Riese, Adam: Rechenbuch (1520, 1574); *113
Robert of Chester: Liber algorismi de numero indorum (1120); 49
Roller, H.: Perspectiva (1546); 372
Rudolff, C: Die Coss (1525); 135
Ruffini, Paola: Delia insolubilita delle equazioni algebraiche generali di grado superiore
al quarto (1803 - 13); 290
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Russell, B., and A. N. Whitehead: Principia mathematica (1910, 1912, 1913); 218
Sanctorius, S.: Medicina statica (1614, 1728); *624
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Seki, Kova: Kai Fukudai no Ho (1683); 639
Stevin, Simon: De Beghinselen der Weegcoonst (1586); *600
Stifel, Michael: Arithmetica integra (1544); *134, 150
: Die Coss (1553); *299
Stbr, L.: Geometria et perspectiva (1556); *373, *396
Tartaglia, Niccolo: General tractato di numeri et misure; ^114
Taylor, Brook: Methodus incrementorum directa et inversa (1715); *768, *769
Taylor, Richard and Andrew Wiles: Article in Annals of Mathematics (May 1995); 334
Trenchant, Jean: Arithmetique (1557); 146
Veblen, 0., and J. H. C. Whitehead: Foundations of Differential Geometry (1932); 370
Vesalius, A.: De humani corporis fabrica (1543); 96
Viete, Francois: In artem analyticem isagoge (1591); 299
- : Variorum de rebus mathematicis responsorum liber VIII (1593); 93
- : De aequationum recognitione et emendatione (1615); 319
Vlacq, Adrian, and E. de Decker: Het tweede deel van de Nieuwe telkonst (1627); 152
- : Arithmetica logarithmica (1628); 104, 152
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- : De algebra tractatus (1685); 87
Walther, Hans: Proverbia sententiaeque latinitatis ; (Gottingen, 1963); 17
Warren, John: A Treatise on the Geometrical Representation of the Square Roots of
Negative Numbers (1828); 600
Wessel, Caspar: Om Directionens analytiske Betegning (1797); 88
White, E. B.: Charlotte's Web. HarperCollins Publishers (1952); *781
White, E. E.: A Complete Artithmetic. Cincinnati: Van Antwerp, Brigg & Co. (1870); *124
Whitehead, J. H. C. and O. Veblen: Foundations of Differential Geometry (1932); 370
Widmann, J.: Behennde unnd hiipsche Rechnug auff alien Kauffmannschaften (1489);
*103
Wiener, N.: "The Historical Background of Harmonic Analysis" (1938); extract cited
from Felix E. Browder: "Mathematics and the Sciences", in Aspray,
William, and Philip Kitcher (editors): History and Philosophy of Modern
Mathematics. Minneapolis: University of Minnesota Press (1988); 909
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Xylander, G.: Arithmetica (1575); 108
Zenoderus: On Isometric Figures (c. 180 B.C.); 670

Name Index
ABEL,N.H, 132,300,776
ACHILLES, 275
ADELARD of Bath, 463
AGNESI, M., 680
AGRIPPA, C, 205
AHMES, the Scribe, 98
ALBERTI, L., 367
ALEMBERT, J. L. R. d'
harmonic analysis, 909
ratio test, 277
al-KASHANI, J. M., see under K
al-KASHI, J. M., see under K
al-KHOWARIZMI, see under K
Angstrom, a. j., 54
apollonios
analytic geometry, 548
conic sections, 408
helix, 795
history of geometry, 365
APPEL,K, 204
ARBOGAST, L.-F.-A., 686, 1016
ARCHIMEDES
analytic geometry, 548
equations, 316
fractions, 38
history of calculus, 674
history of geometry, 365
large numbers, 29
n, 90,479
spiral, 579
trisection of angle, 424
ARCHYTAS, 422
ARGAND, J. R., 88,600
ARISTARCHUS, 462
ARISTOTLE
history of geometry, 365
logic, 216
ARYABATHA, 91,460
BAILLY,J. S., 780
BARROW, I.
Euclid's Elements, 416
history of calculus, 675
BERNOULLI, Daniel
combinatorics, 184
harmonic analysis, 909
BERNOULLI, Jakob
Bernoulli equation, 998
brachistochrone problem, 590
catenary, 538
history of calculus, 679
infinity symbol, 30
BERNOULLI, Jakob (continued)
lemniscate, 581
logarithmic spiral, 581
probability, 963
BERNOULLI, Johann
Bernoulli equation, 998
brachistochrone, 590
history of calculus, 679
L'Hospital's rule, 782
BERNOULLI, Nikolaus
combinatorics, 184
infinity symbol, 30
BETTI, E., 369
BEVERLEY, William, 209
BEZOUT,E., 304
BHASKARA, 91,289
Pythagorean theorem, 436
BINET, J.-P.-M., 288
BOLYAI, J., 369, 382
BOMBELLI, R.
imaginary numbers, 87
powers, 134
radical notations, 135
BOOLE, G., 216, 252
BORREL,J., 108
BOSE, Raj Chandra, 195
BOUGUER,P., 109
BRAHE,T., 152,894
BRAHMAGUPTA
equations, 298
negative numbers, 72
71, 91
BRAILLE, L., 66
BRIANCHON, C. J., 434
BRIGGS,H., 152
BRING, E.S., 300
BROUWER, L. E. J., 370
BUHLER,J., 333
BtJRGI,J., 153
BUTEO,J., 108
CAESAR, G. J., 18
CAMERARIUS, J., 425
CANTOR, G., 232,257
Cantor set, 630
CARDANO, G.
equations, 299, 316
imaginary numbers, 87
negative numbers, 73, 87
probability, 963
radical notations, 135
CARROLL, L., 227
CARTESIUS, see DESCARTES

1054
NAME INDEX
CAUCHY, A. L.
determinants, 639
functions, 345
history of calculus, 680
mean-value theorem, 718, 719
ratio test, 277
CAVALIERI, B.
history of calculus, 674
theorem of, 456
CAYLEY, A., 202, 204, 639
CELSIUS, A., 466
CEULEN, L. van, 92
CHAPPE, C, 68
CHEBYSHEV, P. L., 964, 977
CHENG TA-WEI, 208
CHESTER, Robert of, 49
CHUQUET, 135
CHU SHIH-CHIEH, 141
CLAUSEN, T., 94
COETS, H., 413
COHEN, P.J, 262
COPERNICUS, N, 464, 894
COTES, R, 581
COURANT, R, 936
CRAMER, G, 639,668
CRANDALL, R, 333
CRELLE,A. L, 301
CYRIL, Saint, 12
d'ALEMBERT, see ALEMBERT
DASE,Z, 94
DECKER, E. de, 152
DEDEKIND, J. W. R, 70, 258
de FERMAT, see FERMAT
DEFOE, D, 32
DELAMAIN, R, 176, 177
de LAPLACE, see LAPLACE
de la VALLEE-POUSSIN,
see VALLEE-POUSSIN
de L'HOSPITAL, G. F, see HOSPITAL
de LUNESCHLOS, J, 296
de MOIVRE, A.
complex numbers, 791, 793
normal distribution, 979
probability, 964
de MORGAN, A, see MORGAN, A. de
DESARGUES,G, 367,434
DESCARTES, R.
amicable numbers, 83
analytic geometry, 548
Cartesian product, 241
equations, 299
history of calculus, 675
history of geometry, 367,368,369
logarithmic spiral, 581
polyhedron formula, 448
DESCARTES R. {continued)
powers, 134
product notation, 104
ad-DIN, Naser, 463
DIOPHANTOS, 134,298,330
DIRICHLET, P. G. L, 336, 910
DODGSON, C. L, 227
DOUADY, A, 634
DURER,A, 206
EBBE,A, 2
EINSTEIN, A, 369, 383, 904, 964
ENGSTROM, A, 30
EPEE, C. M, d', 67
EPIMINEDES, 219
ERATOSTHENES
circumference of the Earth, 462, 467
prime numbers, 77
ESCHER,M. C, 375
EUCLID of Alexandria
common notations, 381
equations, 297
Euclidean geometry, 381
history of calculus, 674
history of geometry, 365, 367
Mersenne prime, 82
postulates, 381
prime numbers, 78
QED, 159
reductio ad absurdum, 159
regular polygons, 415
EUCLID of Megara, 365
EUDOXUS (EUDOXOS), 89, 110
history of calculus, 674
history of geometry, 365
EULER, L.
amicable numbers, 83
calculus of variations, 681
combinatorics, 184
constant, 273
diagrams, logic reasoning, 177
e, 85
Euler diagram, 177
Euler's formula, 507, 802
Fermat numbers, 78
Fermat's last theorem, 334
functions, 336
graph theory, 184,221
harmonic analysis, 909
history of calculus, 680,681
history of geometry, 369
history of trigonometry, 465
Konigsberg bridge problem, 202
magic square, 209
multiplier, Euler's, 997
perfect numbers, 82
71, 86,93

NAME INDEX
1055
EULER, L. (continued)
polyhedron formula, 448
powers, 134
prime numbers, 78
prime numbers, 78
quintic equation, 300
summation notation, 105
zeta function, 778
EVCLEIDIS,
see EUCLID of Alexandria
FAULHABER, J., 212
FEISTHAMEL, 209
FERGUSON, D.F., 94
FERMAT, P. de
amicable numbers, 83
analytic geometry, 548
combinatorics, 184
Fermat numbers, 78
history of calculus, 675
last theorem, 333
parabolic spiral, 581
prime numbers, 78
probability, 963
spiral of, 581
FERRARI, L., 299,320
FERRO, S. del, 316
FIBONACCI, L.
equations, 299
Fibonacci numbers, 286
Hindu-Arabic numerals, 50
71, 91
prime numbers, 77
trigonometry, 464
FINCKE, T., 461, 465
law of tangents, 486
FISHER, R. A., 964
FONTANA, Niccolo, see TARTAGLIA
FOURIER, J. B. J., 336
harmonic analysis, 909
FRANKLIN, B., 208
FRANKLIN, Ph., 204
FREDHOLM, E. I., 682
FREGE, G., 216, 235
GAFURIUS, F., 922
GALILEI, G.
combinatorics, 184
history of calculus, 674
introduction to transfinite numbers, 257
motion, 893
GALOIS, E., 301
GALTON, F., 979
GAULLAUDET, T. H., 67
GAUSS, C. F.
complex numbers, 88,
GAUSS, C. F. (continued)
determinants, 639
Fermat's last theorem, 333
fundamental theorem of algebra,
299,300
large numbers, 30
Latin, 19
modular arithmetic, 123
non-Euclidean geometry, 369, 382
normal distribution, 979
prime number theorem, 80
regular heptadecagon, 417
regular polygons, 417
row operations, 661
statistics, 964
vector history, 600
GEMMA FRISIUS, R., 116
GERARD of Cremona, 461,463
GERMAIN, S., 334,681
GHALIGAI, 135
GIBBS, J. W., 601
GODEL,K, 160,262
GOLDBACH, C, 81
GRASSMANN, G., 601
GREGORIE, J., 93, 301, 768
GREGORIUS XIII, 18
GREGORY XIII, 18
GULDIN, P., 866, 884
GUNTER,E., 176,461
GUTHRIE, F., 203
HADAMARD, J. S., 80, 778
HAKEN, W., 204
HAMILTON, W. R.
Hamilton paths, 203
laws of algebra, 133
matrices, 639
quaternions, 601
vector history, 600
HARDY, G. H., 778
HARRIOT, T., 104, 109, 134
HASPER, W., 39
HAUY, R.-J., 399
HEAWOOD, P. J., 204
HEIN,P., 571
HEISENBERG, W.K, 906
HERIGONE, 134
HERMITE, Ch.
e, 85,
matrices, 658
HERODIAN, A., 37
HERODIANUS, A., 37
HERON
cube roots, 148
equations, 316
formula of, 444

1056
NAME INDEX
HILBERT, D. , 201,
history, mathematics, 301, 369
Konigsberg, 201
23 important problems, 369
Peano curve, 635
HIPPARCHOS, 90, 366, 462
HIPPOCRATES of Chios, 422
HOBBES, T., 3
HODDER, 116
H0EG,P., 70,370
HOLZMANN,W., 108
HOOKE, R., 869
HORACE, 17
HOSPITAL, G. F. de L'
brachistochrone problem, 590
history of calculus, 679
I/Hospital's rule, 782
HUBBARD, J., 634
HUILIER, S. L', 780
HULSIUS, L., 466
HUTTON, Ch., 673
HUYGENS, C.
catenary, 538
cycloid, isochronous property, 591
history of calculus, 678
probability, 963
HYPATIA, 366
ISIDORE of Seville, 5
JACOBI, C. G. J., 639
JAENISCH, 209
JODE (JUDAEIS), C, 490
JOHN of Seville, 463
JONES, W., 85
JORDAN, C.
history of geometry, 369
Jordan curve, 379
row operations, 662
JOULE, J. P., 867
JULIA, G. M., 633
JUSTINIAN I, 40
JUSTINIANUS, P. S. F., 40
KANT, I., 201
al-KASHANI, J. M., 91,238
al-KASHI, J. M., 91,320
KASNER, E., 29
KEMPE,A. B., 204
KEPLER, J.
combinatorial geometry, 184
Guldin's first rule, 866
history of calculus, 674
logarithms, 152
motion, 894
Platonic solids, 398
semi-regular polyhedra, 399
volume, wine barrels, 872
al-KHOWARIZMI, 48, 100, 298
al-KHOWARIZMI, 48, 100, 298
KIRKHAM, T. P, 198
KLEIN, F.
classification of geometries, 382
Erlangen program, 370
Klein bottle, 380
KOCH, N. H. F. von, 627
KOLMOGOROV, A. N., 965
KOPERNIK, M., see COPERNICUS
KOVALEVSKI, S., 682
KRAMP, Ch., 106
KRONECKER, L., 71
KYRILLOS, 12
LABOCHEVSKI, N. I., 369, 382
LAGRANGE, J. L.
derivative, notation of, 686
determinants, 639
figurate numbers, 291
history of calculus, 680
mean-value theorem, 716
multipliers, 873
partial derivative, notation of, 710
prime number theorem, 80
probability, 964
sixtic equation, 300
variation of constants, 1025
LAMBERT, J. H.
history of geometry, 367,369
hyperbolic functions, 534
LAME, G., 127, 334
LAPLACE, P. S. de
determinants, 639, 648
history of calculus, 680
normal distribution, 979
probability, 964
LEGANDRE, A. M., 336
Fermat's last theorem, 334
prime number theorem, 80
LEHMNER, D. H., 334
LEHMNER, E., 334
LEIBNIZ, G. W. von
binary numeration, 62
brachistochrone problem, 590
combinatorics, 184
derivative, notation of, 686
determinants, 639
functions, 336
fundamental theorem of calculus,
748
history of calculus, 676
history of geometry, 369
integral, notation of, 722
lingua universalis, 216
logic reasoning, use of diagrams,
242
mathematical logic, 216, 217

NAME INDEX
1057
LEIBNIZ, G. W. von {continued)
multiplication sign, 104
negative distance, concept of, 548
negative numbers, 73
7i, 93
LEONARDO DA VINCI, 399
LEONARDO of Pisa, see FIBONACCI
L'HOSPITAL, G. F. de, see HOSPITAL
L'HUILIER, S., 780
LIBBY, W. F., 1002
LIE, M. S., 370, 682
LINCOLN, A., 21
LINDEMANN, F. von, 86
LINNE, C. von, 5
LIOUVILLE, J., 85,
LISTING, J. B., 369
LIU HUI, 91
LOBACHEVSKI, N., 369
LORENTZ, H. A., 904
LUCAS, F. E. A., 79,287
LUNESCHLOS, J. de, 296
LYAPUNOV, A. M., 964
MACHIN,J., 94
MACLAURIN, C, 770
MANDELBROT, B., 626
Mandelbrot set, 633
MARKOV, A. A., 964
MAUPERTUIS, P.-L., 466
MEIR EZRA, Ibrahim ben, 185
MENDOZA, Antonio de, 60
MENELAUS, 463
MENGER, K, 632
MERCATOR, G., 596
MERSENNE, M., 79
METHODIOS, Saint, 12
MILLES, C, 892
MITTAG-LEFFLER, G., 262, 682
MOBIUS,A., 369,380
MOIVRE, see de MOIVRE
MONGE, G., 367
MORGAN, A. de
algebra of sets
de Morgan's laws, 250
four-color map problem, 204
proof by induction, 158
MORGENSTERN, 0., 965
MORSE, S. F. B., 66
NAPIER, J.,
logarithms, 150 et seq.
Napier's bones, 174
NEHEMIAH, the Rabbi, 95
NEUMANN, J. von, 218, 301, 965, 968
NEWTON, I.
binomial expansion, 776
brachistochrone problem, 590
NEWTON, I. (continued)
fundamental theorem of calculus, 748
history of calculus, 676, 677, 678
history of geometry, 369
motion, 894,895
negative distance, concept of, 548
numerical analysis, 301
numerical solution, 941
powers, 134
NEYMAN, J., 965
NICHOLAUS the Fifth, 41
NICOMACHOSofGerasa, 82,110
NOETHER, E., 301
NUMA (Roman king), 17
NUNES,P., 596
NUNEZ SALACIENCE, P., 596
OTHO,V., 92,465
OUGHTRED, W., 104, 109, 110
slide rule, 176
PACIOLI, L., 102,135, 399, 418
PAGANINI, N., 83
PAO CHHI-SHOU, 214
PAPPUS (PAPPOS)
history of calculus, 674
history of geometry, 366
Pappus's and Guldin's theorem, 866,
884
PARKER, E. T., 195
PARMENIDES, 218
PASCAL, B.
combinatorics, 184
history of calculus, 675
Pascal's theorem, 434
Pascal's triangle, 141
probability, 963
PEANO, G.
axioms, 157
Peano curve, 635
e , 233
PEARSON, E., 965
PEARSON, K, 964
PEET, T. E., 99
PELL, J., 288
PENROSE, L. S., 374
PENROSE, R., 374
PITISCUS, B., 458, 459, 465
PLATO (PLATON)
Achademya, 364
calculus, 674
Platonic solids, 398
PLATO of Tivoli, 463
POINCARfi, J. H., 158, 369, 382
calculus, 682
POISSON, S.-D., 964,979
PONCELET, J. V., 367,434
PRISCUS,T., 18

1058
NAME INDEX
PROCLUS, 366
PTOLEMY (PTOLEMAEUS)
history of geometry, 366,367
history of trigonometry, 463
law of sines, 480
motion, 894
theorem of, 402
PYTHAGORAS
equations, 297
history of calculus, 674
music, 922
Pythagorean theorem, 435
reductio ad absurdum, 159
QUETELET, A., 979
RAHN, J. H., 105, 135
RECORDE, R.,
abacus, 170
addition, 116
equality sign, 107
multiplication, 118
plus and minus notations, 103
REGIOMONTANUS, J., 103, 464
REISCH, G., 115,116, 218, 870,890, 922,
RENARD, Ch., 929
REUTERSVARD, 0., 374
RH/ETICUS, G. J., 464
RHIND, A.H., 98
RICHMOND, H.W., 417
RIEMANN, G. F. B.
definite integral, 747, 748
history of geometry, 369
non-Euclidean geometry, 383
zeta function, 778
RIENGEL, G., 204
RIESE,A., 113
ROBERT of Chester, 49, 463
ROBERVAL, G.P. de, 590, 675
RODLER, H., 372
ROGET, P. M., 177
ROLLE, M., 716
ROOMEN, A. van, 92
RUDOLFF, C, 135
RUFFINI, P., 300
RUSSELL, B., 218,235
RUTHERFORD, E., 964
SACCHERI, G., 369
SANCTORIUS, S., 624
SARRUS,J. P., 647
SCHEDEL, H., 41
SCHWARTZ, F., 682
SEKIKOWA, 639
SERVOIS, F.-J., 133
SHANKS,W., 94
SHARP, A., 94
SHRIRKHANDE, S. S., 195
SIERPINSKI, W., 631
SIMPSON, T., 955
SIMSON, R., 287
SMITH, L. R., 94
SOSIGENES, 18
STEVIN, S., 134,600
STIFEL, M., 134,135,150, 299
STIRLING, J., 770
STOR, L., 373,396
STORM P. (PETERSEN), 337, 354
STRABO (STRABON), 467
STRINDBERG, A., 682
SYLVESTER, J. J., 639
TARSKI, A., 160
TARTAGLIA, N., 114, 316
TAYLOR, B., 768
TAYLOR, R., 334
THALES
history of trigonometry, 461
theorem of, 430
THOMAS of Sarzana, 41
TORRICELLI, E., 674
TRENCHANT, J., 146
TSU CH'UNG-CHIH, 91
TURING, A.M., 218,301
ULAM, S.M., 968
VALLEE-POUSSIN, C. J., 80, 778
VANDERMONDE, A.-T., 639
VARIGNON, P., 580
VEBLEN,0., 370,379
VENN, J., 242
VESALIUS,A., 96
VIETE (VIETA), F.
equations, 299, 316
law of cosines, 483
7i, 92,93
powers, 134
VINOGRADOV, I. M., 81
VLACQ, A.
logarithms, 152
multiplication sign, 104
radical sign, 135
VOGELIN,J., 425
von NEUMANN, J., 218, 301, 965, 968
al-WAFA, Abu, 463
WALLIS, J.
complex numbers, 87
history of calculus, 675
infinity symbol, 30
7i, 93
powers, 134
WALTHER, H., 17

WARREN, J., 600
WEIERSTRASS, K. T. W., 682
WENZELIDES, 209
WESSEL, C, 88,600
WHITEHEAD, A. N., 218
WHITEHEAD, J. H. C, 370
WIDMANN, J., 103
WIENER, N., 301,909
WILES, A., 333
WOLFRAM, S., 301
WREN, Chr., 590
WRENCH, J. W., Jr., 94
XYLANDER, 108
YANGHUI, 213
YOUNGS, J. W. T., 204
YU, the Emperor, 205
ZENO (ZENON), 275, 674, 893
ZENODORUS, 680

1060
Subject Index
An indexed page number might be the first page of a subject
and further information could be found on following pages.
A, 54
a, an, one, etymology of, 27
abacus, 168
abelian group, 132
abscissa, 472, 549
absolute
convergence, 280
error, 161
inequality, 312
maximum, 341, 804
minimum, 341, 804
absolute value/values, 71
addition/subtraction, laws, 133
division laws, 133
equations, 308
law, 132
multiplication laws, 133
absorption laws, Boolean algebra, 254
abstraction, geometric, 370
abundant numbers, 82
acceleration, 897
average, 897
free fall, 611
instantaneous, 897
integration, 900
vectors, 901
accuracy, 50, 161
Achilles and the tortoise, 275
acre, 409
acute
angle, 388
triangle, 393
addend, 115
addition
complex numbers, 789
fractions, 125
fundamental operations, 115
matrices, 642
parallelogram law, 604
polynomials, 129
symbols of, 102
triangle law, 604
vectors, 603
adjacent angles, 388
adjoint matrix, 658
aggregation symbols, 122
Ahmes papyrus, 98
Akkadian numeration, 36
aleph-null, 258
Ale's Stones, 468
algebra
Boolean, 252
coefficients, 114
constants, 114
definition, 100, 114
fundamental theorem, 299, 305
matrices, 642
sets, 249
transfinite numbers, 261
variables, 114
vectors, 602
algebraic
expression, 128
functions, 348
numbers, 84
vector, 641
algorithm, Euclidean, 126
alphabet
Cyrillic, 12
Etruscan, 12
Greek, 11
Latin, 12
Phoenician, 10
phonetic, 10
alphameric, 54
alphanumeric, 54
alternate
exterior angles, 389
exterior angles, theorem, 426
interior angles, 389
interior angles, theorem, 426
alternating series
approximation of sum, 283
convergence, 280
altitude, see height
ambiguous case, 480
amicable numbers, 83

SUBJECT INDEX
1
amplitude, polar coordinate system, 576
analytic geometry, 547
history, 548
angle/angles
adjacent, 388
bisector, Euclidean construction, 414
central, 429
chord, circle, 431
chord/tangent, circle, 431
complementary, trigonometric functions,
471
definitions, 388,389
depression, 494
elevation, 494
equal, Euclidean construction, 413
orthogonal coordinate system, 472
peripheral, 429
plane, measurements, 410
polar, coordinate system, 576
reentrant, 392
rotation, 592
salient, 392
sight, 494
60°, Euclidean construction, 415
solid, measurements, 410
supplementary, 388
supplementary, trigonometric functions,
474
theorems, 426
triangle, external, 428
triangle, opposite, 428
trisection of, 423
vectorial, 576
angstrom, A, 54
antecedent, 110,222
anti-commutative property, vector product,
619
antiderivative, 722
anti-snowflake curve, 629
anti-square curve, 629
apex, cones, 402
approximate
integration, 947
results, 161
values, calculation with, 162
approximated values, 161
approximation
Fourier series, 916
methods of, 923
remainder of series, 281
sums of alternating series, 281
sums of series, 281
Arabic
numerals, 6
numeration, ancient, 36
numeration system, 48
arbitrary constant, integration, 722
arc
circle, 390
cosecant, 477
cosine, 477
cotangent, 477
definition, 387
length, circle, 440
length, method of calculus, 827
length, parametric form, 832
length, polar coordinate system, 835
network, 202
secant, 477
sine, 477
tangent, 477
arccos, 477
derivative, 697
arccosec, 477
arccot, 477
derivative, 697
arccsc, 477
derivative, 697
arcctg, 477
arch, 539
arcosech, 539
derivative, 704
arcosh, 539
derivative, 704
arcoth, 539
derivative, 704
arcsch, 539
arcsec, 477
derivative, 697
arcsin, 477
derivative, 697
arctan, 477
derivative, 697
arcth, 539
arcus function, 477
area/areas
between two curves, 853
circle, 853
curvilinear figures, 442
cyclic quadrilateral, 445
definite integral, 749
ellipse, 852
methods of calculus, 851
polar coordinates, 856
quadrilateral, 445
quadrilateral, cyclic, 445

1062
SUBJECT INDEX
area/areas {continued)
rectilinear figures, 441
snowflake curve, 627
surface of revolution, 860
torus, surface, 866
triangle, 489, 552, 653
triangle, determinants, 653
triangle, trigonometric method, 489
units, measurements, 409
argument, 336,788
polar coordinate system, 576
Aristotelian logic, 216
arithmetic
definition, 100, 113
four simple rules, 113
mean, 975
modular, 123
residue classes, 123, 137
series, finite, 266
arm, moment, 838
armillary sphere, 463
arsech, 539
derivative, 704
arsh, 539
arsinh, 539
derivative, 704
artanh, 539
derivative, 704
arth, 539
as, 16
associative laws, 132, 249, 254
Boolean algebra, 254
sets, 249
astroid, 594
equations, parametric, 594
perimeter, 833
Attic numeration, 36
atto, 54
augmented matrix, 662
auxiliary equation, 1018
average, 975
acceleration, 897
value, 751
velocity, 897
axiom/axioms
definition 157
Peano, 157
axis/axes
Earth, 491
ellipse, 390
imaginary, number plane, 788
polar, 576
real, number plane, 788
axis/axes (continued)
rotation, 403
solid of rotation, 403
azimuth, 576
Aztec numeration, 60
Babylonian sexagesimal system, 56
barber paradox, 236
barycenter, 838
base/bases
hanging, logarithms, 156
cone, 402
cylinder, 401
power, 136
prism, 400
trapezium, 394
trapezoid, 394
triangle, 393
basic integration formulas, 724
bearings, 491, 499
Bernoulli equation, 998
bifurcation point, 348
billion, U.S., 28
binary, 46
counting rules, basic, 62
numeration, 62
Binet formula, 288
binomial/binomials, 128
coefficient, 196
distribution, 969
expansion, power series, 775
linearly dependent, 663
linearly independent, 663
theorem, 140
biquadratic equations, 311
birectangular, 404
bis, 16
bisection method, 935
flowchart, 940
bisector, 389
angle, Euclidean construction, 414
triangle, 439
bit/bits, 54
blocks of elements, systems, 198
Boolean
algebra, 252
constants, 220
expressions, simplifying, 254
Borromean rings, 786
bottle, Klein, 380

SUBJECT INDEX
1063
boundary estimation
Simpson's rule, 959
trapezoidal rule, 952
boundary values, calculation with, 166
boundedness laws, Boolean algebra, 253
braces, 122
brachistochrone problem, 590
brackets, 122
Braille, 66
Branchion's theorem, 434
bridges, Konigsberg, 201
Briggs's logarithms, 155
broken line, 387
buck, 43
byte/bytes, 54
c, continuum, 262
calculations, reliability, 161
calculators, trigonometric values, 476
calculus
differential, definition, 673
fundamental theorem, 748
history, 674
infinitesimal, 673
integral, 673,721
of variations, definition, 673, 681
calendar
Gregorian, 18
Julian, 18
Mayan, 59
Roman Republican, 18
ten-month, 17
cancellation laws, 133
Cantor set, fractals 630, 636
dimension, 636
cap, spherical, 403
capacitance, 1011
cardinal
numbers, 5
points, 499
cardioid, 859
carpet, Sierpinski, 631
Cartesian coordinates system, 549
casting out nines, 123
casus irreducibilis, 319
catenary, 538,831
length, 831
catenoid, 538
cat's cradle, 370
Cauchy's mean-value theorem, 718
Cavalieri's theorem, 446
C-bill, 43
celestial sphere, 463
Celtic languages, 21
center
of circle, Euclidean construction, 413
of convergence, 767
of mass, 838,845
of mass, solids of revolution, 845
of sphere, 403
centesimal degree, 410
centi, 54
central
angle, 391
tendency, 974
centroid/centroids, 837
see also center of mass
lamina, formula, 845
CGPM, 53
chain rule
derivatives, composite functions, 691
of inference, 227
changing
base, logarithms, 156
limits, integration, 750
chaos, 348
characteristic, 155
equation, 1018
Euler's, 448
Chebyshev's theorem, 977
chessboard
geometric series, 270
magic squares, 208
Chinese
mercantile numerals, 45
numerals, traditional, 44
numeration, 44
official numerals, 45
rod numerals, 51
chord, 387
circle, 391
sphere, 403
theorem, 438
chromic number, 203
cipher, 26
circle/circles, 406
arc length, 440
area, 442,853
center, Euclidean construction, 413
circumference, 440
definition, 390, 559
equations, 559, 568, 586
equations, parametric, 586
great, 403
magic, 213
parallel, torus, 405
polar equation, 578
sector, area, 442
segment, area, 442
small, 403
squaring, 422

1064
circuits, logic, 229
equivalent switch, 230
circumcenter, triangle, 432
circumference, definition, 390
circumscribed circles, radii, 440
Classical Greek number names, 10, 13
classical logic, 216
closed
curve, 379
half-line, 387
interval, 111
C-note, 46
Codex Mendoza, 60
coefficient/coefficients
algebra, 114
binomial, 196
determinant, 669
Fourier, 910
matrix, 662
cofactor/cofactors, 648
inverse of a matrix, 659
cofunction identities, 507
collinear, 434
column
matrix, 640
vector, 641
column index
determinant, 646
matrix, 640
combinations, 196
combinatorial geometry, 184
combinatorics, 183
history, 184
comma, decimal, 75
common
difference, series, 264
divisor, 303
factor, 303
fractions, 74
logarithms, 155
notations, Euclidean, 381
ratio, series, 264
common denominator
greatest, 126
least, 125
commutative laws, 132, 249, 252
Boolean algebra, 252
sets, 249
commutative properties, vectors, 615, 619
complement, 240
complementary angles, trigonometric
functions, 471
complementary solution, 1024
SUBJECT INDEX
complement laws, Boolean algebra, 253
completing the square, 310
integration, 726
complex
conjugate matrix, 658
matrix, 658
number plane, 87, 708
complex numbers, 71, 87, 131, 787
conjugate, 131
logarithms of, 794
polar form, 788
powers of, 791
principal logarithm of, 795
roots of, 793
components
ordered, 241
vector product, 618
vectors, 608
composite functions, 345
derivatives, 691
magic squares, 210
numbers, 77
compound-angle identities, 509
compound interest, 1003
concave
downward, 798,801
polygon, 392
polyhedron, 396
upward, 798,801
conclusion, 222,227
concurrent vertices, 434
condition, Dirichelet's, 910
conditional
convergence, 280
equation, 303
inequality, 109,312
probability, 966
cone/cones, 402
formulas, 452
volume, 447
congruence, 376
congruent triangles, 427
conies, 406,408,559
vertex equation, 568
conic sections, 406, 559
conjecture/conjectures
Goldbach, 81
Kepler's, 184
conjugate
binomial surds, 139
complex numbers, 131
conjunctions, 221
consequent, 110

SUBJECT INDEX
1065
constant/constants
algebra, 114
Boolean, 220
Euler's, 273
integration of, 722
Ludolphine, 93
proportionality, 349
spring, 1008
term, polynomial, 128
variation of, method, 1025
constant-coefficient linear
differential equations
homogeneous, 1018
nonhomogeneous, 1024
equations, homogeneous, 1018
constraint, functions, 822
construction, Euclidean, 413
continued fractions, 127
root extraction, 143
continuity and limits, 355
continuous
dilution, 1004
function, 355
continuum hypothesis, 262
contraction, length, 904
contradiction, principle of, 216
contrapositive, 223
convergence, 270
absolute, 280
alternating series, 280
center of, 766
comparison test, 276
conditional, 280
infinite geometric series, 274
interval of, 766
n-th term test, 270
power series, 766
ratio test, 277
selected series, 271
Simpson's rule, 959
trapezoidal rule, 952
converse, 223
conversion/conversions
fractions, 75, 125
half-angle tangent, 512
integrals, trigonometric /algebraic, 740
orthogonal/polar coordinates, 577
products, sums of functions, 513
spherical/orthogonal coordinates, 584
conversion table
hyperbolic functions, 545
trigonometric functions, 475
convex
polygon, 392
polyhedron, 396
cooling, 1005
coordinate geometry, 548
coordinate system/systems
Cartesian, 549
cylindrical, 582
orthogonal, 472,549
orthogonal/polar, transformation, 887
polar, arc length, 835
polar, plane, 576
polar, space, 582
rectangular, 549
right-handed, 607
spherical, 583
vectors, 607
correction factor, inversion, 648
correspondence, 376
corresponding angles, 389, 426
cosecans hyperbolicus, 534
cosecant, 470
curve, 502
derivative, 695
hyperbolic, 534
cosech, derivative, 702
cosh, derivative, 702
cosine/cosines, 470
curve, 503
derivative, 695
hyperbolic, 534
law of, 483
of x, power series expansion, 773
cosinus hyperbolicus, 534
cotangens hyperbolicus, 534
cotangent, 470
curve, 504
derivative, 695
hyperbolic, 534
coth, derivative, 702
counting
numbers, 70
rules, binary, basic, 62
counting boards, 168
courses, 491
Cramer's rule, 668
critical points, 798, 799, 816
crossing lines, 387
crunode, 386
cube/cubes, 397,400,448
duplication, 421
formulas, 450
magic, 214
twin cube, 455
volume, 446
cubic
equations, 316
meter, 409
numbers, 292

1066
cuneiform symbols, 35
cup, 238
curve/curves, 387
anti-snowflake, 629
anti-square, 629
closed, 379
cosecant, 502
cosine, 503
cotangent, 504
family of, 992
Jordan, 379
length, curvilinear figures, 440
Peano, 635
secant, 503
sine, 502
snowflake, 627, 636
snowflake, dimension, 636
tangent, 504
curvilinear figures, curve length, 440
curvilinear motion, 896
cusp/cusps, 386, 590
cybernetics, 909
cyclic
permutations, 192
quadrilateral, area, 445
cycloid/cycloids, 590
equations, parametric, 591
cylinder/cylinders
formulas, 452
surface, chromic number, 204
volume, 446
cylindrical coordinates, 582
Cyrillic alphabet, 12
d'Alambert's ratio test, 277
damped oscillations, 1009
damping constant/resistance, 1011
Danish number names, 24
dating, radiocarbon, 1002
deca, 54
decadic, 46
decagon, 392
regular, Euclidean construction, 416
decay, radioactive, 1001
deceleration, 897
December, 19
deci, 54
decimal
comma, 75
fractions, 75
marker, 75
point, 75
position system, 46, 48
system of measurement, 52
SUBJECT INDEX
deduction, proof by, 158
defective numbers, 82
deficient numbers, 82
definite integral, 722, 747
area, 749
deformation, 379
degree/degrees
angle, 410
centesimal, 410
cyclic permutation, 192
differential equation, 992
polynomial, 128
Delian problem, 421
deltoid, 395
de Moivre's
formula, 791
laws, 250,254
laws, Boolean algebra, 254
denary, 46
denominator, 74
greatest common, 126
least common, 125
rationalizing, 139
density, probability, function, 971
denumerably infinite sets, 258
dependence, linear, 1017
dependent variable, 336
depression, angle of, 494
derivative/derivatives, 685
algebraic functions, 689
arccos, 695
arccot, 695
arccsc, 695
arcosech, 704
arcosh, 704
arcoth, 704
arcsec, 695
arcsin, 695
arctan, 695
arsech, 704
arsinh, 704
artanh, 704
composite functions, 691
cosecant, 695
cosech, 702
cosh, 702
cosine, 695
cotangent, 695
coth, 702
exponential functions, 699
functions, algebraic, 689

derivative/derivatives {continued)
functions, composite, 691
functions, exponential, 699
functions, logarithmic, 699
functions, transcendental, 695
fundamental trigonometric functions
695
inflection point, 798
inverse trigonometric functions, 697
linear functions, 686
logarithmic functions, 699
maximum, 798
minimum, 798
mixed partial, 712
nonlinear functions, 687
partial, mixed, 712
power of a variable, 689
product, 690
quotient, 691
secant, 695
sech, 702
sine, 695
sinh, 702
sum, 690
tangent, 695
tanh, 702
transcendental functions, 695
descent, path, most rapid, 590
detachment, principle of, 223
determinant, coefficient, 669
determinants, 637, 646
history, 638
properties, 649
determinate
forms, limits, 361
straight line, 387
deviation, standard, 976
diagonal/diagonals, 392
diagonals/sides, parallelogram, 438
face, 396
main, determinant, 647
main, matrix, 641
matrix, 641
principal, determinant, 646
principal, matrix, 641
secondary, determinant, 646
space, 396
diagram
Euler, 242
Venn, 242
diameter, 391
sphere, 403
SUBJECT INDEX 1067
difference, 116
approximate values, 163
boundary values, 166
common, series, 264
complex numbers, 130, 789
indeterminate, 785
sets, 239
differentiability, 693
differential calculus, 673, 683
differential equations, 989
first-order ordinary, 994
harmonic, 1022
homogeneous, 995
linear, 1016
ordinary, 991,994
partial, 991
second-order ordinary, 1015
differentials, 685
differentiation
implicit, 705
logarithmic, 705
partial, 710
digamma, 37
digits, 5
reliability, 161
reliable, 162
significant, 50
valid, 164
digon, 404
dilation of time, 905
dilution, continuous, 1004
dimension/dimensions
concept, 376, 635
definitions, 376,635
fractals, 636
Hausdorff - Besicovich, 636
matrix, 640
one, 376
similarity, 636
three, 376
two, 376
zero, 376
Diophantine equations, 330
direct
proof, symbolic logic, 227
proportionality, 349
directrix, 401
conic sections, 406
Dirichelet's condition, 910
discontinuous functions, 355
expansion, Fourier series, 915
discriminant, 311, 319

1068
SUBJECT INDEX
disjunctions, 222
disk method, volume, 873
dispersion, 976
distance, 550, 896, 1090
formulas, 550
distribution
binomial, 969
Gaussian, 979
normal, 979
standard normal, 983, 985
theory, 682
distributive laws, 132
Boolean algebra, 253
sets, 250
distributive properties, vectors, 615
dividend, 120
divine section, 418
divisibility, rules of, 121
division
complex numbers, 790
fractions, 125
fundamental operations, 120
polynomials, 129
symbols of, 104
divisor, 120
common, 303
doctrine of opposites, 291
dodecahedron, 397, 398, 448
formulas, 451
rhombic, 399
domains
functions, 339,342,505
trigonometric functions, 505
double-angle identities, 510
double integral, 753, 754, 887
transforming, coordinates, 887
double saw, 43
dual, 3
elements, 387
elements, projective geometry, 434
operations, 387
theorem, projective geometry, 434
duality principle
Boolean algebra, 253
projective geometry, 434
duodecimal numeration, 46, 61
duodenary numeration, 46
duplation, Egyptian method, 118
duplication of the cube, 421
Dutch number names, 25
dyadic, 46
e, 85
e1*, power series expansion, 792
e*, power series expansion, 774
East Arabic numerals, 49
eccentricity, 406
echelon form, system of linear equations,
661
edge, polyhedron, 396
Egyptian
numeration, 34
papyri, 98
triangle, 393
eine messbare Unendlichkeit, 30
Einstein's energy-mass relation, 906
electric
charge, 1011
current, 1011
electromotive force, 1011
element/elements
blocks of, systems, 198
determinant, 646
leading, 640
matrix, 640
series, 264
set, 232
elementary
functions, 348
geometry, definition, 376, 386
row operations, 661
elevation, angle of, 494
eleven, etymology, 27
ellipse, 406
area, 442,852
definition, 390,559
equations, 560,568,587
equations, parametric, 587
perimeter, 440
super-, 571
ellipsoid, 405
of revolution, volume, 876
oblate, 405
prolate, 405
reflectors, 431
elliptic
geometry, 383
helix, 597
empty sets, identities, 250
endpoint extrema, 804
energy/mass, 904
energy-mass relation, Einstein's, 906
English number names, etymology of, 26
entry
determinant, 646
leading, 640
matrix, 640
pivot, 662
epicycles, 894
epicycloids, 592
equations, parametric, 592
Epiminides' paradox, 219

SUBJECT INDEX
1069
equality
sign, Rhind Papyrus, 99
symbols of, 99, 107
equation/equations
absolute value, 308
astroid, parametric, 594
auxiliary, 1018
Bernoulli, 998
biquadratic, 311
characteristic, 1018
circle, 559,568,586
circle, parametric, 586
conditional, 303
cubic, 316
cycloid, parametric, 591
definition, 302
differential, 989
differential, first-order ordinary, 994
differential, harmonic, 1022
differential, homogeneous, 995
differential, ordinary, 991, 994
differential, partial, 991
differential, second-order ordinary, 1015
Diophantine, 330
ellipse, 560, 568
ellipse, parametric, 587
epicycloid, parametric, 592
exponential, 314
fractional, 307,309
functional, 354
general form, 305
helix, right elliptic, 597
history, 297,316
homogeneous, 328
hyperbola, 565,568,588
hyperbola, parametric, 588
hypocycloid, parametric, 593
intersecting lines, 556
linear, 307
linear, systems, solution, 661
linear differential, 1016
logarithmic, 315
monic form, 305
normal, 688
parabola, 563,568,589
parabola, parametric, 589
parallel lines, 555
parametric, 586
parametric, area, surface of revolution,
864
parametric, helix, conical, 596
parametric, helix, cylindrical., 595
parametric, helix, spherical, 596
perpendicular lines, 555
equation/equations {continued)
point-slope, 553
polar, 578
polynomial, 307
quadratic, 309
quartic, 320
quintic, 300
reduced, 1017
root, 313
root/coefficient relationships, 350
slope-intercept, 553
solution, iterative methods, 935, 941
solutions, graphic methods, 931
standard form, 305
straight lines, 553
symmetric, 329
systems of, 325
systems of, graphic solution, 933
tangent, 688
terminology, 302
theory of, 295
trigonometric, 516
two-point, 553
types of, 302
equiangular spiral, 580
equilateral triangle, 392
Euclidean construction, 415
equilibrium, vector forces, 610
equivalence
logical, 223,224
topological, 369,378
equivalent switch circuits, 230
Erlangen program, 370
error/errors, 161
absolute, 161
estimate, Taylor's series, 770
relative, 161
rounding, 165
trapezoidal rule, 952
type I, 970
type II, 970
Etruscan
alphabet, 12
number names, 15
etymology
a, an, one, 27
eleven, 27
English number names, 26
hundred, 27
million, 28
thousand, 27
twelve, 27

1070
SUBJECT INDEX
Euclidean
algorithm, 126
common notations, 381
construction, 413
geometry, 381
postulates, 381
Euler
diagram, 242
paths, 203
Euler's
characteristic, 448
constant, 273
Euler's formula, 507, 792
line, 433
multiplier, 997
polyhedron formula, 448
product, 778
even functions, expansion, 918
even/odd functions, 340
even-odd identities, 508
exa, 54
excess, spherical, 404
excluded middle, principle of, 216
existential quantifier, 221
expansion
determinants, Laplace, 648
functions, even, 918
functions, odd, 919
explicit form, function, 339
exponent, power, 136
exponential
equations, 314
form, fundamental trigonometric
functions, 470
functions, 353
functions, derivatives, 699
expression/expressions
algebraic, 128
Boolean, simplifying, 254
extended mean-value theorem, 718
exterior angles, 389
external force/electromotive force, 1011
extraction, roots, 141
extraneous solutions, trigonometric
equations, 523
extrema and critical points, 797
extreme/extrema
endpoint, 804
functions, 341
relative, 799
two or more independent variables, 815
extremes, 111
face
diagonal, 396
lateral, prism, 400
lateral, pyramid, 400
factor/factors, 303
common, 303
integrating, 996
prime, 303
theorem, 304
factorial
symbol, 106
tree, 186
factoring, 303
quadratic equations, 309
trigonometric equations, 519
fall through resisting medium, 1007
family of curves, 992
femto, 54
Fermat
numbers, 78
prime-testing method, 80
Fermat's
last theorem, 333
spiral, 581
Fibonacci
Association, 288
numbers, 287
sequence, 286
field, 71
fifteen puzzle, 193
figurate numbers, 289
four dimensions, 293
higher dimension, 294
finite
decimal fractions, 76
series, 266
sets, 234
first minor, 648
first-order ordinary differential equations,
994
flat angle, 388
flow chart
bisection method, 940
Newton's method, 946
fluid ounce, 409
focus
conic sections, 406
ellipse, 390
focused perspective, 372
foot, 409
U.S., definition, 52
force, 611,896
external, as electromotive force, 1011
vectors, 605
forced oscillations, 1010

SUBJECT INDEX
1071
formula/formulas
see also theorem
basic integration, 724
Binet, 288
boundary, trapezoidal rule, 952
center of mass, solids of revolution, 845
centroid, lamina, 845
de Moivre's, 791
distance, 550
Euler's, 507,792
Heron's, 444
Maclaurin's, 769
quadratic, 311
quadrature, 947
Taylor's, 767
four-color map theorem (problem), 203
Fourier
coefficients, 910
series, 910
four simple rules of arithmetic, 113
fractals, 625
fraction/fractions, 5, 74
Archimedes' system, 38
basic counting rules, 125
common, 74
continued, 127
continued, root extraction, 143
conversion, 75, 76, 125
decimal, 75
Egyptian numeration, 34
improper
Latin number names, 16
partial, 130
proper, 74
Roman numeral system, 42
sexagesimal, 57
unit, 126
fractional equation, 307, 309
Fraktur, 19
free oscillations, 1008
French number names, 20, 24
friction coefficient/resistance, 1011
friendly numbers, 82
frustum
cone, 402
pyramid, 400
full angle, 388
function/functions
algebraic, 348
algebraic, derivatives, 689
arcus, 477
calculations with, 343
composite, 345
function/functions (continued)
constraint, 822
continuous, 355
definition, 336
discontinuous, 355
discontinuous expansion, Fourier series,
915
distribution, normal, 979
elementary, 348
even, expansion, 918
even/odd, 340
explicit form, 339
exponential, 353
exponential, derivatives, 699
half-angle, 512
history, 336
hyperbolic, 533
hyperbolic, derivatives, 702
hyperbolic, inverse, 539
hyperbolic, inverse, derivatives, 704
implicit form, 339
inverse hyperbolic, derivatives, 704
inverses, 346
linear, 349
logarithmic, 354
logarithmic, derivatives, 699
normal distribution, 979
odd, expansion, 919
odd/even, 340
one-to-one, 346
polynomial, 348
power, 350
power, integration, 723
probability, density, 971
quadratic, 351
rational, 348
restrictions, 822
Riemann zeta, 778
transcendental, 349
transcendental, derivatives, 695
transcendental, expansion, 771
trigonometric, fundamental, 470
trigonometric, inverse, 477
functional equation, 354
fundamental hyperbolic functions, graphs,
537
fundamental theorem of algebra, 299, 305
fundamental theorem of arithmetic, 78
fundamental theorem of calculus, 748
fundamental trigonometric functions, 470
derivatives, 695
exponential form, 470
fundamental trigonometric identities, 507

1072
G, grand, 43
gallon, 409
game
fifteen puzzle, 193
Saint Peter's, 184
theory, 965
Gaussian
distribution, 979
elimination, 661
reduction, 661
Gauss-Jordan
elimination, 662
reduction, 662
GB, 55
GCD, 126
General Conference,Weights/Measures, 53
general form
of algebraic polynomial equation, 305
of equation for line, 555
generalized me an-value theorem, 718
general power rule, 692
general solution, differential equation, 992
general term, series, 264
general theory of relativity, 904
generator, 401
generatrix, 401
genus, surface, 379
geocentric system, 894
geodesic, 491
geoid, 491
geometric
abstraction, 370
mean, 111,268
series, finite, 268
geometry/geometries
analytic, 547
analytic, history, 548
combinatorial, 184
coordinate, 548
elementary, definition, 376, 386
elliptic, 383
Euclidean, 381
Euclidean, definition, 376
history, 364
hyperbolic, 382
infinitude of, 384
non-Euclidean, 376, 381
parabolic, 382
plane, definition, 376
Riemann, 383
solid, definition, 37
types of, 377
SUBJECT INDEX
Germanic languages, 25
German number names, 25
giga, 54
gigabyte, 55
gnomons, 291
Gobar, 50
Goldbach conjectures, 81
golden
number, 287
ratio, 418
rectangle, 419
section, 418
triangle, 419
gon, 410
googol, 29
googolplex, 29
grand, 43
graph, network, 202
graphical interpolation, 927
graphs
fundamental hyperbolic functions, 537
inverse hyperbolic functions, 539, 540
trigonometric functions, 502
graph theory, 184,201
great circle, 403
greater than, symbol, 109
greatest common denominator, 126
Greek, 10
alphabet, 11
number names, 10
numeration, 36
transliteration, 12
Greenlandic number names, 7
Greenwich meridian, 491
Gregorian calendar, 18
grid, polar, 577
grouping notations, 122
group theory, 301
growth, population, 1000
Guldin's first rule, 866
Guldin's second rule, 884
half-angle, functions, 512
half-cone, 402
half-life, radioactive isotopes, 1001
half-line, 387
half-open interval, 111
half-saw, 43
Hamilton paths, 203
Hanoi, tower of, 285

SUBJECT INDEX
1073
harmonic
analysis, 907
differential equations, 1022
oscillations, 1008
sequence, 264
series, 264
series, finite, 266
Hausdorff - Besicovich dimension, 636
heating, 1005
Hebrew, ancient, numeration, 36
hecto, 54
height
cone, 402
prism, 400
trapezium, 394
trapezoid, 394
triangle, 393,439
zone, sphere, 403
Heisenberg's uncertainty principle, 906
helix
conical, 596
conical, equations, parametric, 596
cylindrical, 595
cylindrical, equations, parametric, 595
elliptic, 597
spherical, 596
spherical, equations, parametric, 596
hemisphere, 403
heptadecagon, regular, Euclidean
construction, 417
heptagon, definition, 392
Hermitian conjugate matrix, 658
Herodianic numeration, 36
Heron's formula, 444
heuristic method, mathematics, teaching,
370
hexadecimal numeration, 46, 64
hexagon, 392
hexagram/hexagrams, 420
magic, 213
hexahedron, 397
Hilbert curve, see Peano curve, 635
Hindu-Arabic numeration system, 48
homeomorphism, 369, 378
homogeneous
constant-coefficient linear differential
equations, 1018
differential equations, 995
equations, 328
linear differential equations, 1016, 1017
Hooke'slaw, 869
hundred, etymology, 27
hyperbola,
definition, 406, 564
equations, 565, 568, 588
equations, parametric, 588
hyperbolic
cosecant, 534
cosine, 534
cotangent, 534
functions, 533
functions, derivatives, 702
functions, inverse, 539
geometry, 382
secant, 534
sine, 534
spiral, 580
tangent, 534
hypocycloid/hypocyclods, 593
equations, parametric, 593
hypotenuse, 393
hypothesis, 157,222
continuum, 262
null, 970
Riemann, 778
i, imaginary unit, 71, 87
i1, properties, 793
Icelandic number names, 25
icosahedron, 397, 448
formulas, 451
idempotent laws, Boolean algebra, 253
identity/identities
addition , 72
cofunction, 507
compound-angle, 509
double-angle, 510
empty sets, 250
equations, 303
even-odd, 508
fundamental hyperbolic functions, 541
fundamental trigonometric, 507
half-angle, 512
hyperbolic, 541
inverse hyperbolic functions, 544
inverse trigonometric functions, 514
laws, 132
laws, Boolean algebra, 252
matrix, 641
multiple-angle, 510
multiplication, 72
odd-even, 508
principle of, 216
Pythagorean, 508, 541
symbol, 108
trigonometric, 507

1074
SUBJECT INDEX
imaginary
axis, number plane, 788
numbers, logarithms, 156
numbers, pure, 71
unit, 71,87
implications, 222
implicit
differentiation, 705
form, function, 339
impossible figures, 374
improper
fraction, 74
integral, 756
subset, 237
inch, 409
inclusion theorem, 528
incompleteness theorem, 160
indefinite
continuation of straight line, postulate,
381
integral, 722
independence, linear, 1017
independent variable, 336
indeterminate
differences, 785
forms, limits, 360, 779
powers, 785
products, 785
quotients, 784
index
column, determinant, 646
column, matrix, 649
root, 138
row, determinant, 646
row, matrix, 649
indirect proof, 159
symbolic logic, 227
Indo-European languages, 9
inductance, 1011
induction, proof by, 158
inequality/inequalities, 312
absolute, 312
conditional, 109, 312
symbols, 108
unconditional, 109, 312
inference, chain rule, 227
infinite
decimal fractions, 76
geometric series, 274
integrands, 756
limits, integrals, 760
series, 270
sets, 234
infinitesimal calculus, 673
infinitesimals, 673
infinity, 29
inflection point/points, 798
derivative, 798
functions, 341
inflection tangent, 388
inscribed circles, radii, 440
instantaneous
acceleration, 897
velocity, 897
integers, 70, 72
integral/integrals, 722
calculus, 673, 721
definite, 722,747
double, 753
equation, 309
improper, 756
indefinite, 722
multiple, 753
primitive, 722
Riemann, 749
triple, 753
unrestricted, 756
integrand/integrands, 722
infinite, 756
integrated mean values, 751
integrating factors, 996
integration
acceleration, 900
approximate, 947
basic formulas, 724
constant, 722
methods of, 723
numerical, 947
partial fractions, 745
by parts, 728
power functions, 723
by power series expansion, 746
recursion formulas, 730
reduction formulas, 730
by substitution, 733
trigonometric identities, 727
velocity, 900
Intercalarius, 18
interest, compound, 1003
interior angles, 389
International System of Units, 52
interpolation, 926
graphical, 927
linear, 926
intersecting lines, equations, 556
intersection, 239
interval/intervals, 111
convergence, 766
invariants, 369

SUBJECT INDEX
1075
inverse, 223
function, 346
matrices, 655
matrix, cofactors, 659
matrix, solution systems, linear
equations, 665
inverse hyperbolic functions, 539
derivatives, 704
graphs, 539,540
inverse proportionality, 350
inverse trigonometric functions, 477
derivatives, 697
identities, 514
inversion correction factor, 648
inversions, 191
invertible matrix, 656
Ionic numeration, 36
Irish number names, 22, 23
irrational numbers, 70, 84
irregular matrix, 656
isosceles triangle, 392
Euclidean construction, 415
Italian number names, 20
iterative methods, solutions, equations
935, 941
J, joule, 867
Jordan curve, 379
joule, J, 867
Julian calendar, 18
Julia set, 633
K, 55
kappa, 37
Kepler's
conjecture, 184
laws, motion, 894
kilo, 54
kilobyte, 55
kites, 395
Klein bottle, 380,204
knight, magic squares, 208
knots, topology 379
Kbnigsberg bridges, 201
Ladin, 21
Lagrange's
mean-value theorem, 716
method of multipliers, 823
multiplier, 823
lamina, centroid, 842
language/languages
Arabic, 4
Austronesian, 4
Celtic, 21
Classical Greek, 4, 10
Danish, 24
language/languages {continued)
English, 26
Etruscan, 15
French, 20,24
German, 25
Germanic, 25
Greek, 4,10
Greenlandic, 4, 7
Icelandic, 25
Indo-European, 9
Intipik, 4, 7
Irish, 22,23
Italian, 20
Ladin, 21
Latin, 14
Maori, 9
Middle English, 4,5,26
Modern English, 4, 26
Old English, 4,5,26
Old Norse, 26
Rhaeto-Romance, 21
Romance, 20
Romanian, 20
Romansh, 21
Sanskrit, 9,26
Scots, 22,23
Spanish, 20
Swedish, 25
Dutch, 25
Welsh, 22,23
Laplace expansion, determinants, 648
large numbers, law of, 967, 977
lateral face
prism, 400
pyramid, 400
lateral surface
cone, 402
cylinder, 4011
Latin
alphabet, 12
language, 14
number names, 14
squares, 195
latitude, 491
law/laws
absolute, 132
absolute values, addition/subtraction, 133
absolute values, division, 133
absolute values, multiplication, 133
absorption, Boolean algebra, 254
additive inverse, 132
arithmetic and algebra, 132
associative, 132
associative, Boolean algebra, 254
associative, sets, 249
boundedness, Boolean algebra, 253
cancellation, 133

1076
law/laws {continued)
commutative, 132
commutative, Boolean algebra, 252
commutative, sets, 249
commutative, vectors, 615
complement, Boolean algebra, 253
cosines, 483, 485
de Morgan's, 250
de Morgan's, Boolean algebra, 254
derived, arithmetic and algebra, 133
distributive, 132
distributive, Boolean algebra, 253
distributive, sets, 250
distributive, vectors, 615
duality, Boolean algebra, 253
Hooke's, 869
idempotent, Boolean algebra, 253
identity, 132
identity, Boolean algebra, 252
Kepler's, motion, 894
large numbers, 967, 977
multiplication, zero, 133
multiplicative inverse, 132
negation, 133
Newton's, motion, 894
parallelogram, vectors, 603
product, zero, 133
reflexive, 132
sines, 480,482
substitution, 132,224
supplementary, Boolean algebra, 253
symmetry, 132
tangents, 486
transitive, 132
triangle, vectors, 603
law of cosines, 483
spherical triangles, 485
law of sines, 480
spherical triangles, 482
law of tangents, 486
layer, spherical, 403
LCD, 125
leading
element, 640
entry, 640
least common denominator, 125
left-hand continuity, 358
legion, 14
legs, triangle, 393
lemma, definition, 157
lemniscate, 581
length
contraction, 904
as unit of measurement, 409
SUBJECT INDEX
less than, symbol, 109
level, significance, 970
L'Hospital's rule, 782
Vhomme moyen, 979
liar paradox, 219
libra, 16
Lie groups, 682
light rays, reflection, conies, 408
limit/limits, 355
calculations with, 360
changing, integration, 750
determinate forms, 361
indeterminate forms, 358, 779
infinite, integrals, 760
infinite series, 270
one-sided, 358
trigonometric functions, 528, 529
undefined forms, 362
line/lines
concurrent, theorems, 432
definitions, 387
equation, general form, 555
equation, standard form, 555
Euler's, 433
geodesic, geodetic, 491
intersecting, equations, 556
parallel, equations, 555
perpendicular, equations, 555
rhumb, 596
segment, midpoint, 551
of sight, 494
straight, equations, 553
straight, polar equations, 578
linear
dependence, 1017
differential equations, 1016
equations, 307
equations, systems, solution, 661
functions, 349
functions, derivatives, 686
independence, 1017
interpolation, 926
linearly
dependent binomials, 663
independent binomials, 663
lingua universalis, 216
liter, 409
lituus, 581
local
maximum, 341
minimum, 341
locus, loci, 387
logarithm/logarithms, 150
base, changing, 156
Briggs's, 155
calculation rules, 154
common, 155

SUBJECT INDEX
1077
logarithm/logarithms (continued)
complex numbers, 794
definitions, 154
history, 150
imaginary numbers, 156
Napier's (Napierian), 155
natural, 155, 354
negative numbers, 156
principal, complex numbers, 795
logarithmic
differentiation, 705
equations, 315
functions, 354
functions, derivatives, 699
spiral, 580
logic
Aristotelian, 216
circuits, 229
classical, 216
mathematical, symbolic 215
predicate, 216
symbolic, 215
logical equivalence, 223
logistic, 100
longitude, 491
Lorentz transformations, 904
lost solutions, trigonometric equations,
lower triangular matrix, 641
loxodrome, 596
Ludolphine constant, 93
Ludus Sancti Petri, 185
lune/lunes
spherical, 404
squaring, 422
lute, Pythagoras's, 420
Maclaurin's series, 769
magic
circles, 213
cubes, 214
hexagrams, 213
pentagrams, 213
spheres, 213
square, 15-by-15, 212
squares, 205
squares, chessboards, 208
squares, composite, 210
magnitude, vectors, 608
main diagonal
determinant, 646
matrix, 641
major axis, ellipse, 390
major premise, 227
Mandelbrot set, 633
mantissa, 155
manual sign language, 67
Maori number names, 9
map, 188,336
mapping, 188,336,378
map theorem (problem), four-color, 203
mass, 611, 896
center, 838
as inductance, 1011
mass/energy relation, 906
mass/time, 905
moment of, 838
mass point systems, 838
mathematical (symbolic) logic, 215
history, 216
mathematical
model, 1011
probability, 966
proof, 157
matrix/matrices, 637
adjoint, 658
augmented, 662
coefficient, 662
complex, 658
complex conjugate, 658
Hermitian conjugate, 658
history, 638
inverse, 655
inverse, solution system of linear
equations, 665
invertible, 656
irregular, 656
nonsingular, 656
normal, 658
orthogonal, 657
projection, 658
singular, 656
skew-Hermitian, 658
skew-symmetric, 654
square, 640
symmetric, 654
transpose, 654
triangular, 641
unitary, 658
maximum
absolute, 804
derivative, 798
Mayan
calendar, 59
numeration, 58
MB, 55
mean
arithmetic, 975
geometric, 111, 268
proportional, 111
values, integrated, 751
weighted, 975

1078
SUBJECT INDEX
means, 111
mean-value theorem/theorems, 716
extended, 718
generalized, 718
restricted, 717
measurements
decimal system of, 52
length, area, volume, 409
median/medians, 974
intersection, triangle, 433
trapezium, trapezoid, 394 , 394
triangle, definition, 393, 439
mega, 54
megabyte, 55
members, set, 232
Menger sponge, 632
Mercator projection, 596
meridian/meridians
Greenwich, 491
of longitude, 491
torus, 405
Mersenne
numbers, 79
primes, 79
messbare Unendlichkeit, 30
meter, 409
Meter Convention, 52
method/methods
Lagrange's, multipliers, 823
Monte Carlo, 968
Newton's, 941
radiocarbon, 1002
Simpson's, 955,957
trapezoidal, 949
undetermined coefficients, 1027
variation of constants, 1025
metric system, 52
micro, 54
microlocal analysis, 682
Middle English number names, 26
midline, see median
midpoint
of line segment, 550
of line segment, Euclidean construction,
413
mile
nautical, 409
statute, 409
milli, 54
milliard, 28
millio, 15,28
million, etymology, 28
minimal surface, 538
minimum, absolute, 804
minor
first, 648
principal, 648
minor axis, ellipse, 390
minor premise, 227
minuend, 116
minus symbol, Rhind Papyrus, 99
minute, angle, 410
mixed surds, 139
Mobius strip, 380,204
mode, 974
modular arithmetic, 123
modulus, complex numbers, 788
moment arm, 838
moment of mass, 838
lamina, 844
monad, 13
monic form (equation), 305
monomial, 128
monotonicity, functions, 341
Monte Carlo methods, 968
Morse code, 66
Moscow Papyrus, 99
most rapid descent, path, 590
motion, 891
curvilinear, 896
Kepler's laws, 894
Newton's laws, 894
oscillating, 1008
rectilinear, 896
multiple, scalar, 602
multiple-angle identities, 510
multiple integrals, 753
multiplicand, 118
multiplication
complex numbers, 790
Egyptian, 118
fractions, 125
fundamental operations, 118
inverse, law, 132
matrices, 642
polynomials, 129
principle, 186
Russian peasant, 119
symbols of, 104
table, 118
vectors, 614
multiplicative notation, 44
multiplier/multipliers, 118
Euler's, 997
Lagrange's, 823
music and numbers, 922
myriad, 10,29
N, newton, 605
names, numbers, 7
nano, 54

SUBJECT INDEX
1079
Napierian logarithms, 155
Napier's
bones, 174
logarithms, 155
rods, 174
nappe/nappes, 402
natural
numbers, 70,72
oscillations, 1008
natural logarithms, 155, 354
power series expansion, 773
navigation, 490
negation, 220
laws, 133
negative numbers, 72
logarithms of, 156
negligible terms, method of approximation,
925
networks, 201
logic circuits, 229
newton, N, 605
Newton's
laws, motion, 894
method, 941
method, flow chart, 946
rc-gon, definition, 392
nihil, 15
nine-point triangle, 433
nines, casting out, 123
node, 386
nonagon, 392
nonary, 46
non-denumerably infinite sets, 260
non-Euclidean
geometries, 381
postulates, 381
nonhomogeneous
constant-coefficient linear differential
equations, 1024
linear differential equations, 1016, 1017
non-integrability, 725
nonlinear functions, derivatives, 687
nonsingular matrix, 656
nontrivial zeros, 778
normal
distribution, 979
equation, 688
form, equation, 305
matrix, 658
notation/notations
see also symbol
additive, 32,34
additive, Roman, 40
common, 381
multiplicative, 32, 44
place-value, 32
notation/notations (continued)
positional, 32, 46
scientific, 53
subtractive, 32
subtractive, Roman, 40
n-th term test, infinite series, 270
November, 19
null
angle, 388
hypothesis, 970
matrix, 641
vector, 603
vector, matrix, 641
number/numbers
abundant, 82
algebraic, 84
amicable, 83
cardinal, 5
chromic, 203
complex, 71, 87,131, 787
composite, 77
counting, 70
cubic, 292
defective, 82
deficient, 82
Fermat, 78
Fibonacci, 287
figurate, 289
friendly, 82
golden, 287
gnomon, 291
irrational, 70, 84
Mersenne, 79
natural, 70,72
negative, 72
oblong, 291
ordinal, 5
Pell, 288
pentagonal, 291
perfect, 82
preferred, 929
prime, 77
Pythagorean, 332
random, 967
rational, 70,72
real, 70
sets of, 234
square, 290
square pyramidal, 293
supertetrahedral, 294
tetrahedral, 292
transcendental, 85
transfinite, 257
triangular, 289
number mystique, 6

1080
SUBJECT INDEX
number names, 7
Classical Greek, 10, 13
Danish, 24
Dutch, 25
Etruscan, 12
French, 20,24
German, 25
Greek, 10
Greenlandic, 7
Icelandic, 25
Irish, 22,23
Italian 20
Latin, 14
Maori, 9
Middle English, 26
Old English, 26
Old Norse, 26
Rhaeto-Romance languages, 21
Romanian, 20
Sanskrit, 26
Scots, 22,23
Spanish, 20
Swedish, 25
Welsh, 22,23
number plane, complex, 87, 788
numbers, vs. infinity, 29
number theorem, Pythagorean, 332
numerals, 5
Arabic, 6
Chinese rod, 51
East Arabic, 49
Gobar, 50
mercantile Chinese, 45
official Chinese, 45
Roman, 6
traditional Chinese, 44
West Arabic, 49
numeration system/systems, 31
Akkadian, 36
Arabic, 48
Arabic, ancient, 36
Attic, 36
Aztec, 60
binary, 62
duodecimal, 61
Egyptian, 34
Greek, 36
Hebrew, ancient, 36
Herodianic, 36
hexadecimal, 64
Hindu-Arabic, 48
Ionic, 36
Mayan, 58
octal, 64
Roman, 39
sexagesimal, 56
numeration system/systems {continued)
Sumerian, 35
vigesimal, 58
numerator, 74
numerical integration, 947
oblate ellipsoid, 405
oblique
cone, 402
cylinder, 401
prism, 400
oblong numbers, 291
obtuse
angle, 388
triangle, 393
octagon, 392
octahedron, 397,448
formulas, 450
octal, 46
octal numeration, 64
octenary, 46
October, 19
odd/even functions, 340
odd-even identities, 508
odd functions, expansion, 919
Old English number names, 26
Old Norse number names, 26
one-sided
limits, 358
surfaces, 380
one-to-one functions, 346
opart, 375
open
half-line, 387
interval, 111
operational precedence, 122
operator, 336
opposite angles, 389, 426
opposites, doctrine of, 291
optical art, 375
order
determinant, 646
differential equation, 991
matrix, 640
ordered
components, 241
sample, 187
ordinal numbers, 5
ordinary differential equations, 991, 994
first-order, 994
second-order, 1015
ordinate, 472,549
origin, 549
number line, 71
vectors, 602
orthodrome, 596

SUBJECT INDEX
1081
orthogonal
matrix, 657
trajectories, 1012
orthogonal coordinate system, 472, 549
rotation, 573
translation, 572
orthogonal/polar coordinate
transformation, 887
oscillating motion, 1008
oscillations, 1008
damped, 1009
forced, 1010
harmonic, 1008
undamped, 1009
osculation, point of, 386
outcomes, favorable/unfavorable, 966
papyri, Egyptian, 98
Papyrus
Ahmes, 98
Moscow, 99
Rhind, 98,297
parabola, 406, 562
equations, 563, 568, 589
equations, parametric, 589
segment, area, 442
parabolic
geometry, 382
spiral, 581
paraboloid, 405
paraboloid reflectors, 432
paradox
barber, 236
Cretian, 219
Epiminides', 219
liar, 219
Russell's, 235
parallel circles
torus, 405
parallelepiped, 400
volume, 446
parallel line/lines, 387
equations, 555
Euclidean construction, 414
postulate, 381
parallelogram law, vectors, 603
parallels of latitude, 491
parameter, elimination, 597
parametric equations, 586
arc length, 832
area, surface of revolution, 864
parentheses, 122
partial
differential equations, 991
differentiation, 710
fractions, 130
fractions, use of, integration, 745
sums, 271
particular solution, differential equations,
993
Pascal's
theorem, 434
triangle, 140
path, descent, most rapid, 590
paths
Euler, 203
Hamilton, 203
Peano
axioms, 157
curve, 635
Pell number sequence, 288
pendulum property, 591
pentacle, 420
pentadecagon, regular, Euclidean
construction, 416
pentagon, 392
regular, Euclidean construction, 415
pentagonal numbers, 291
pentagram/pentagrams, 420
magic, 213
perfect numbers, 82
perimeter, 390
snowflake curve, 627
periodic decimal fractions, 76
periphery, definition, 390
permutations, 188
all elements distinct, 189
cyclic, 192
identical elements, 193
perpendicular line/lines, 387
equations, 555
Euclidean construction, 414
perspective, focused, 372
peta, 54
phase
rhetorical, 101
symbolic, 101
syncopation, 101
Phoenician alphabet, 10
phonetic alphabet, 10
pi, 7i, 85,89,920
"Indiana", 95
"scriptural", 94
pico, 54
pint, 409
pivot
entry, 662
operation, 662

1082
SUBJECT INDEX
pivoting, matrix, 662
place-value notation, 32
planar, 376
plane angle, definition, 388
plane geometry
definition, 376
theorems, 426
Platonic solids, 397, 398
plus symbol, Rhind Papyrus, 99
point/points
bifurcation, 348
cardinal, 499
concurrent, theorems, 432
critical, 798,799,816
decimal, 75
definitions, 386
of inflection, definition, 388
inflection, functions, 341
of osculation, 386
saddle, 816
salient, 386
stable (iteration of functions), 348
stationary, 798
terminal vectors, 602
point-slope equation, 553
polar angle, 576
polar coordinate system
arc length, 835
area, 856
plane, 576
space, 582
polar equations, 578
polar form, complex numbers, 788
polar grid, 577
pole, coordinate system, 576
polygon/polygons, 392
regular, Euclidean construction, 415
spherical, 404
polyhedron/polyhedra, 396
formula, Euler's, 448
semi-regular, 399
polynomial/polynomials, 128
addition, 129
division, 129
equations, 307
functions, 348
multiplication, 129
prime, 303
root extraction, 144
subtraction, 129
population growth, 1000
position, 896
as electric charge, 1011
system, decimal, 48
vectors, 608
positional notation, 32, 46
postmultiplication, 644
postulate/postulates
definition 157
Euclidean, 381
indefinite continuation of straight line,
381
non-Euclidean, 381
parallel, 381
pound, U.S., definition, 52
power/powers, 134
calculation rules, 136
complex numbers, 791
functions, 350
functions, integrating, 723
history, 134
indeterminate, 785
matrices, 644
variable, derivative, 689
power series, 765
convergence, 766
power series expansion
binomial, 775
cosine of x, 773
eP, 792
e*, 773
integration by, 746
natural logarithms, 773
roots, 776
sine of x, 111
transcendental functions, 771
power sets, 238
precedence, operational, 122
predicate logic, 216
preferred numbers, 929
premise, 157
major, 227
minor, 227
premultiplication, 644
prime/primes, 77
factor, 303
Mersenne, 79
numbers, 77
number theorem, 80, 778
polynomial, 303
pseudo-, 80
testing method of Fermat, 80
primitive integral, 722
principal diagonal
determinant, 646
matrix, 641
principal minor, 648
principal values, trigonometric functions,
506

SUBJECT INDEX
principle
of contradiction, 216
of detachment, 223
duality, Boolean algebra, 253
duality, projective geometry, 434
of excluded middle, 216
of identity, 216
Heisenberg's uncertainty, 906
multiplication, 186
two-value, 220
uncertainty, Heisenberg's, 906
prism/prisms, 400
formulas, 451
height, 400
truncated, 400
volume, 446
probability, 961
conditional, 966
density function, 971
history, 963,979
mathematical, 966
problem
brachistochrone, 590
Delian, 421
four-color map (theorem), 203
product/products, 118
approximate values, 163
boundary values, 166
complex numbers, 130, 790
conversion from sums of functions,
derivative, 690
determinants, 650
Euler, 778
indeterminate, 785
scalar, 614
sets, 241
symbol, 105
tree, 186
trigonometric, breaking up, 514
triple, scalar, 622
triple, vector, 623
vector, 617
projectile, path, 589, 901
projection, 373
Mercator, 396
projection matrix, 658
projective geometry, 377
Mercator projection, 396
prolate ellipsoid, 405
proof/proofs
computer generated, 160, 204, 333
direct, symbolic logic, 227
indirect, 159
indirect, symbolic logic, 227
mathematical, 157
proof by
deduction, 158
induction, 158
reductio ad absurdum, 159
proper
fractions, 74
subset, 237
properties,
commutative, vectors, 615, 619
distributive, vectors, 615
proportion, symbols, 110
proportional, mean, 111
proportionality
constant, 349
direct, 349
inverse, 350
proportionals, 111
protolanguage, 8, 25
propositions, 220
pseudo-primes, 80
Ptolemy's theorem, 438
punctum temporis, 18
pure
imaginary numbers, 71
surds, 139
puzzle, fifteen, 193
pyramidal numbers, square, 293
pyramids, 400
formulas, 451
volume, 447
Pythagoras's lute, 420
Pythagorean
brotherhood, 397
doctrine of opposites, 291
identities, 508
numbers, 332
number theorem, 332
theorem, 435
triples, 332
QED, 159
quadrangle, 394
quadrans, 16
quadrant
astronomical instrument, 463
coordinate system,
quadratic
equations, 309
formula, 311
functions, 351
quadrature formulas, 947
quadrilateral, 392,394
area, 445
about a circle, 438
in a circle, 430
quadrivium, 870
quadrual, 4

1084
SUBJECT INDEX
quantifier
existential, 221
universal, 220
quantity, vector, 602
quart, 409
quartic equations, 320
quaternary, 46
quaternion, 601
quinary, 46
Quinctilis, 19
quinta essentia, 398
quintic equations, 300
quipu, 182
quotient/quotients, 120
approximate values, 163
boundary values, 166
complex numbers, 130, 790
derivative, 691
indeterminate, 784
rad, 410
radian, 410
radical, 138
simplified form, 139
radicand, 138
radioactive decay, 1001
radiocarbon dating, 1002
radius/radii
circle, 391
circumscribed circles, 440
coordinate system, 472
inscribed circles, 440
radius vector, complex number, 788
sphere, 403
vector, 576
random numbers, 967
random variable, 971
range/ranges
functions, 339,342,505
measure of dispersion, 976
trigonometric functions, 505
rank, matrix, 664
ratio
common, series, 264
golden, 418
symbols, 110
test, d'Alambert's, 277
rational
functions, 348
numbers, 70, 72
rationalizing, denominator, 139
ray/rays, 387
real
axis, number plane, 788
numbers, 70
zeros, methods of approximation, 931
rectangle, 395
area, 441
golden, 419
rectangular coordinate system, 549
rectilinear motion, 896
recursion formulas, integration, 730
reduced equation, 1017
reductio ad absurdum, proof by, 159
reduction formulas, integration, 730
reentrant angle, 392
reflection, conies, light rays, 408
reflector
ellipsoid, 431
paraboloid, 432
reflex angle, 388
reflexive law, 132
Regula de Tri, 111
regula falsi, 297
regular
decagon, Euclidean construction, 416
heptadecagon, Euclidean construction,
417
pentadecagon, Euclidean construction,
416
pentagon, Euclidean construction, 415
polygon/polygons, 392
polygons, Euclidean construction, 415
polyhedra, theorems, 448, 450
polyhedron/polyhedra, definitions, 397
pyramid, 400
relative
error, 161
extrema, 799
maximum, 341
minimum, 341
minimum, derivative, 798
relativity
general theory of, 904
special theory of, 904
reliability
calculations, 161
digits, 161
reliable digits, 162
remainder
Taylor series, 770
theorem, 303
replacement, samples with, 199
residue classes, 123
complex numbers, 137
resistance, 1011
resisting medium, fall through, 1007
restricted mean-value theorem, 717
restrictions, functions, 822
resultant, vectors, 603, 605

SUBJECT INDEX
1085
results, approximate, 161
Rhaeto-Romance number names, 21
rhetorical phase, 101
rhetorical style, 298
Rhind Papyrus, 98,297
rhodonea, 856
rhombic dodecahedron, 399
rhomboid, 394
area, 441
rhombus/rhombi, 394
area, 441
rhumb line, 596
Riemann
geometry, 383
hypothesis, 778
integral, 748
sum, 747
zeta function, 778
right
angle, 388
cone, 402
cylinder, 401
prism, 400
triangle, 393
right-hand continuity, 358
right-handed coordinate systems, 607
ring, 71
rod numerals, Chinese, 51
Rolle's theorem, 716
Roman
numerals, 6
numeration, 39
Republican calendar, 18
Romance, 20
Romance languages, 20
Romanian number names, 20
Romansch, 21
root/roots, 135
calculation rules, 138
complex numbers, 793
determining, power series expansion,
776
equation/equations, 302, 313
extraction, 141
history, 135
index, 138
root/coefficient relationships, equations,
305
rotation, orthogonal coordinates, 573
rounding, 164
errors, 165
scientific, 165
shopkeepers', 165
row/rows
matrix, 640
target, 663
vector, 641
row index
determinant, 646
matrix, 640
row operations, elementary, 661
rule/rules
see also law, formula, theorem
chain, derivatives, composite functions,
691
Cramer's, 668
general power, 692
Guldin's first, 866
Guldin's second, 884
L'Hospital's, 782
power, general, 692
Regula de Tri, 111
rounding, 164
Sarrus's, 647
Simpson's, 955,957
of three, 111
trapezoidal, 949
Russell's paradox, 235
saddle point, 816
Saint Peter's game, 184
salient
angle, 392
point, 386
sample, ordered, 187
samples, replacement, 199
san, 37
Sanskrit, 9
number names, 26
Sarrus's rule, 647
satisfy, equation, 302
sawbuck, 43
scalar, 602
matrix, 641, 642
multiple, 602
products, 614
product theorem, 614
triple product, 622
scalene triangle, 392
s'choty, 170
scientific notation, 53
reliable digits, 162
scientific rounding, 165
Scots number names, 22, 23
secans hyperbolicus, 534

1086
SUBJECT INDEX
secant, 387,470
circle, 391
curve, 503
derivative, 695
hyperbolic, 534
secant-tangent theorem, 439
sech, derivative of, 702
secondary diagonal, 646
seconds, angle, 410
section
divine, 418
golden, 418
sector
circle, 391
circle, area, 442
spherical, 404
seed, Newton's method, 941
segment
circle, 391
circle, area, 442
segment, line/lines, 387
equal, Euclidean construction, 413
graphs, network, 202
midpoint, Euclidean construction, 413
spherical, 403
self-similar, fractals, 626
semaphores, 68
semilune, 404
semi-regular polyhedra, 399
semis, 16
senary, 46
September, 19
septenary, 46
sequence, 263
see also series
Fibonacci, 286
Lucas, 287
Pell number, 288
series, 263
alternating, 264
alternating, convergence, 280
arithmetic, 264
convergent, 265
divergent, 265
finite, 265,266
finite arithmetic, 266
finite geometric, 268
Fourier, 910
geometric, 264
harmonic, finite, 266
infinite, 265,270
infinite geometric, 274
Maclaurin's, 769
power, 765
power, convergence, 766
Taylor's, 767
set/sets
algebra, 249
builder, 235
Cantor, fractals, 630, 636
contents, 233
denumerably infinite, 258
finite, 234
infinite, 234
Julia, 633
Mandelbrot, 633
non-denumerably infinite, 260
power, 238
product, 241
theory, 231
sexagesimal numeration, 46, 56
sextans, 16
Sextilis, 19
shell method, volume, 877
SI, 52
side condition, functions, 822
Sierpinski
carpet, 631,636
carpet, dimension, 636
sponge, 631,636
sponge, dimension, 636
triangle, 631,636
triangle, dimension, 636
sieve of Eratosthenes, 77
sight
angle of, 494
line of, 494
sighting angle, 494
sign, see symbol
significance level, 970
significant digits, 50
sign language, manual, 67
similar
terms , 128
triangles, 427
similarity, 376
dimension, 636
simplified form, radical, 139
simplifying Boolean expressions, 254
Simpson's rule, 955, 957
boundary estimation, 959
convergence, 959
sine, 470
derivative, 695
hyperbolic, 534
sine curve, 502
sine of x, power series expansion, 771
sines, law of, 480
singular matrix, 656
sinh, derivative, 702
sinus hyperbolicus, 534
size, permutation, 188

SUBJECT INDEX
1087
skew-Hermitian matrix, 658
skew lines, 387
skew-symmetric matrix, 654
slide rule, 176
calculations, 180
construction, 178
history, 176
slope-intercept equation, 553
small circle, 403
snowflake curve, 627, 636
dimension, 636
soap film, 538
soccer ball, 399
solid angles
measurements, 411
solid geometry, 376
theorems, 446
solids
known cross section, volume, 885
Platonic, 397,398
of revolution, 403
solution/solutions, complementary, 1024
extraneous, trigonometric equations, 523
lost, trigonometric equations, 525
soroban, 169
space diagonal, 396
space-time, 904
Spanish number names, 20
special theory of relativity, 904
speed, 897
sphere, 403
armillary, 463
celestial, 463
formulas, 452
magic, 213
surface area, 863
spherical
cap, 403
coordinates, 583
excess, 404
layer, 403
lune, 404
polygon, 404
sector, 404
segment, 403
segments, formulas, 452
triangle, 404
triangles, formula, 453
triangles, law of cosines, 485
triangles, law of sines, 482
wedge, 404
zone, 403
spherical/orthogonal coordinates,
conversions, 584
spheroid, 405
spiral/spirals
Archimedes', 579
equiangular, 580
hyperbolic, 580
logarithmic, 580
parabolic, 581
plane, 579
three-dimensional, 595
sponge
Menger, 632
Sierpinski, 632
spring constant, 1008
as 1/capacitance, 1011
square/squares, 395
area, 441
Euclidean construction, 415
Latin, 195
magic, 205
matrix, 640
meter, 409
numbers, 290
pyramidal numbers, 293
squaring
the circle, 422
lunes, 422
sr, 412
stable point (iteration of functions), 348
standard deviation, 976
form, equation, 305
form, equation, line, 555
normal distribution, 983
Star of David, 420
statements, 220
stationary points, 798, 799
statistics, 962
steradian, 412
stochastic phenomena, 962
straight
angle, 388
line, 387
lines, equations, 553
lines, polar equations, 578
strategy, 965
style,
rhetorical, 298
syncopated, 298
suan-pan, 169
subset/subsets, 237
substitution
integration by, 733
law, 132,224
symbol, integration, 749
trigonometrical functions/algebraic
expressions, 738

1088
SUBJECT INDEX
subtraction
complex numbers, 789
fractions, 125
fundamental operations, 116
matrices, 642
polynomials, 129
symbols of, 102
vectors, 606
subtractive notation, 32
Roman, 40
subtrahend, 116
sum/sums, 115
angles of triangle, 428
approximate values, 163
boundary values, 166
complex numbers, 130, 789
derivative, 690
finite arithmetic series, 267
functions, conversion to products, 513
geometric finite series, 268
infinite geometric series, 274
partial, 271
Riemann, 747
vectors, 603
Sumerian numeration, 35
summation symbol, 105
super-ellipse, 571
supertetrahedral numbers, 294
supplementary angles, 388
trigonometric functions, 474
supplementary laws, Boolean algebra,
surds, 139
surface area
minimal, 538
torus, 866
surface of revolution, 860
surfaces, one-sided, 380
surveying, 490
Swedish number names, 25
syllogisms, 227
symbol/symbols
addition, 102
aggregation, 122
cuneiform, 35
division, 104
equality, 107
factorial, 106
greater than, 109
grouping, 122
identity, 108
inequality, 108
less than, 109
multiplication, 104
operation, 101
symbol/symbols {continued)
product, 105
proportion, 110
ratio, 110
relation, 101
subtraction, 102
summation, 105
symbolic
logic, 215
phase, 101
symmetric
equations, 329
matrix, 654
symmetry, law of, 132
syncopated style, 298
syncopation phase, 101
system, geocentric, 894
Systeme International d'Unites, 53
systems of
blocks of elements, 198
equations, 325
equations, graphic solution, 933
trigonometric equations, 526
table
standard normal distribution, 985
trigonometric values, 476
truth, 220
tangens hyperbolicus, 534
tangent, 388,470
curve, 504
derivative, 695
equation, 688
hyperbolic, 534
sphere, 403
trigonometric function, 470
tangents, law of, 486
tanh, derivative, 702
target row, 663
tautologies, 225
Taylor's series, 767
tendency, central, 974
tera, 54
term
absolute, 309
general, series, 264
linear, 309
quadratic, 309
terminal point, vectors, 602
terminating decimal fractions, 76

terms
negligible, method of approximation,
series, 264
similar, 128
ternary, 46
tessellation, 395
test, d'Alambert's, 277
tetragon, 392,394
tetrahedral numbers, 292
tetrahedron/tetrahedra, 397, 448
formulas, 450
tetraktys, 289
theorem/theorems, 157
see also formula, law, rule
of algebra, fundamental, 299, 305
of arithmetic, fundamental, 78
binomial, 140
Branchion's , 434
Cauchy's mean-value, 718
Cavalieri's, 446
Chebyshev's, 977
chord, 438
concurrent lines, 432
concurrent points, 432
cones, 452
cube, 450
cylinders, 452
de Moivre's, 791
Desargues's, 434
diagonals/sides, parallelogram, 438
dodecahedron, 451
dual, projective geometry, 434
Euler's characteristic, 448
Euler's polyhedron formula, 448
factor, 304
Fermat's last, 333
fundamental, arithmetic, 78
fundamental, calculus, 748
Heron's formula, 444
icosahedron, 451
inclusion, 528
incompleteness, 160
Lagrange's mean-value, 716
lines, concurrent, 432
mean-value, 716
mean-value, Cauchy's, 718
mean-value, extended, 718
mean-value, generalized, 718
mean-value, Lagrange's, 716
mean-value, restricted, 717
octahedron, 450
Pascal's, 434
plane geometry, 426
points, concurrent, 432
SUBJECT INDEX 1089
theorem/theorems (continued)
925 prime number, 80, 778
prisms, 451
Ptolemy's, 438
pyramids, 451
Pythagorean, 435
Pythagorean number, 332
regular polyhedra, 448
remainder, 303
Rolle's, 716
scalar product, 614
secant, 439
secant-tangent, 439
solid geometry, 446
spheres, 452
spherical segments, 452
spherical triangles, 453
tetrahedron, 450
Thales', 430
torus, 453
Vinogradov's, 81
theory,
of games, 965
group, 301
of relativity, general 904
of relativity, special, 904
of sets, 231
thousand, etymology, 27
tiling, 395, 397
time dilation, 905
topological
equivalence, 369, 378
transformation, 369, 378
topology, 377,378
torus, 405
formula, 453
surface, chromic number
volume, 884
tower of Hanoi, 285
trace, matrix, 641
trajectory, 1012
transcendental functions, 349
derivatives, 695
expansion, 771
transcendental numbers, 85
transfinite numbers, 257
algebra of, 261
transformation
synonym of function, 336
topological, 369,378
transformations, Lorentz, 904
transforming
double integrals, 887
orthogonal/polar coordinates, 887
transitive, law, 132

1090
SUBJECT INDEX
translation
axes, volume, 882
orthogonal coordinates, 572
transpose matrix, 654
transposition
cyclic permutation, 192
transversal, 387
trapezium, 394
area, 442
trapezoid, 394
area, 442
trapezoidal rule, 949
boundary estimation, 952
boundary formula, 952
convergence, 952
tree diagram, 186
trial, 4
triangle/triangles, 392
area, 441, 552
area, determinants, 653
area, Heron's formula, 444
area, trigonometric method, 489
circumcenter, 432
congruent, 427
Egyptian, 393
equilateral, Euclidean construction, 415
golden, 419
height, 393,439
isosceles, Euclidean construction, 415
nine-point, 433
Pascal's, 140
Sierpinski, 631
similar, 427
spherical, 404
theorems, 427
types of, 392,393
triangle law, vectors, 603
triangular
matrix, 641
numbers, 289
triens, 16
trifolium, 858
trigonometric
products, breaking up, 514
values, calculators, 476
values, table, 476
trigonometric equations, 516
systems of, 526
trigonometric functions
complementary angles, 471
conversion table, 475
fundamental, derivatives, 695
graphs, 502
inverse, derivatives, 697
principal values, 506
substituting for algebraic expressions, 738
trigonometric functions (continued)
supplementary angles, 474
values, table, 476
trigonometric identities, 507
integration, 727
trigonometry, 457
history, 458,461,467
trinomial, 128
triple/triples
integral, 753
products, 622
Pythagorean, 332
trirectangular, 404
trisection of an angle, 423
trivium, 870
Troy-town, 546
truncation, 164
truth
table, 220
value, 220
tuple, 188
twelve, etymology, 27
twelve-color map problem, 204
twin
cube, 455
primes, 81
two-point
equation, 553
two-value principle, 220
type I error, 970
type II error, 970
uncertainty principle, Heisenberg's, 906
uncia, 16
unconditional inequality, 109, 312
undamped oscillations, 1009
undefined forms, limits, 362
undenary, 46
undetermined coefficients, method of, 1027
Unendlichkeit, eine messbare 30
union, 238
unit
distance, 71
fractions, 126
imaginary, 71, 87
matrix, 641
vectors, 607
vectors, scalar products, 615
unitary matrix, 658
universal set, 235
universal quantifier, 220
universe, 235
unknowns, 114
unrestricted integral, 756
upper triangular matrix, 641

SUBJECT INDEX
1091
valence, 202
valid digits, 164
value/values
absolute, 71
approximated, 161
average, 751
boundary, 166
mean, integrated, 751
principal, trigonometric, 506
trigonometric functions, table, 476
truth, 220
variable/variables
algebra, 114
dependent, 336
independent, 336
polynomial, 128
power, derivative, 689
random, 971
variance, 976
variation of constants, method of, 1025
variations, calculus of, 673, 681
vector/vectors
acceleration, 901
algebraic, 641
components, 608
matrix, 641
product, components, 618
products, 617
quantity, 602
triple product, 623
velocity, 901
vector analysis, 599
history, 600
vectorial angle, 576
velocity, 896
average, 897
as electric current, 1011
instantaneous, 897
integration, 900
Venn diagram, 242
vertex/vertices
angle, 388
concurrent, 434
cones, 402
ellipse, 390
graphs, network, 202
vertical angles, 389
theorem, 426
vibrations, see oscillations
vicenary, 46
vigesimal numeration, 46, 58
Vinogradov's theorem, 81
volume
elementary geometry, 446
ellipsoid of revolution, 876
generated by area bounded by two curves,
880
methods of calculus, 871
shell method, 877
solids, known cross section, 885
torus, 884
translation of axes, 882
units, measurements, 409
wedge, spherical
weight, 611, 896
weighted mean, 975
Welsh number names, 22, 23
West Arabic numerals, 49
work, 867
yard, 409
yocto, 54
yotta, 54
zepto, 54
zero/zeros
angle, 388
Babylonian, 57
dimension, 376
etymology, 26
identity for addition, 72
multiplication law, 133
nontrivial, 778
part of integers, 70,72
product law, 133
real, methods of approximation, 931
value, matrices, predicting, 651
vector, 603
zeta function, 778
zetta, 54
zone, spherical, 403
z score, 978

1092
Symbols in Common Use
Notations and symbols are listed in the order of their
presentation in the text; only those in current use are
included.
Symbol
oo
a IU
£■ ; alb
Page
30
74
. Or , (decimal marker) 75
e
71
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Re (z)
Im (z)
+
—
± ; +
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x or •
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z
n
j
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&
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def
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>
<
>
<
»
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^^
~ or oc
(,); [,];[,[;(,];
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mod
z*9 z
aP
V"; V~~
85
85
87
87
87
103
103
104
104
104
105
105
105
106
108
108
108
108
108
109
109
109
109
109
109
110
111
[,) 111-12
122
123
131
134
135
Symbol
log a
lg
In
lb
QED
p . n p . pn .
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Q
Q.
Q-
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Page
154
154
154
154
159
188
196
220
220
221
221
221
222
222
224
233
233
233
233
233
234
234
234
234
234
234
234
234
234
234
234
Symbol
R
R+
R_
R*
C
c*
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c
2,^
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u
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Page
234
234
234
234
234
234
235
236
236
236
236
237
238
238
238
239
239
239
240
241
241
252
252
252
258
262
270
336
345
346

SYMBOLS IN COMMON USE
1093
Symbol
rad
sr (12)
—
<w
sin
cos
tan
cot
sec
CSC
arcsin
arccos
arctan
arccot
arcsec
arccsc
sinh
cosh
tanh
coth
sech
cosech
arsinh
arcosh
artanh
arcoth
arsech
arcsch
a
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Page
410
412
427
427
470
470
470
470
470
470
477
477
477
477
477
477
534
534
534
534
534
534
539
539
539
539
539
539
602
602
607
607
614
617
640
641
641
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Page
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a
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a
S
S
646
646
654
655
658
658
658
685
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686
686
686
686
710
710
712
722
747
749
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